Surface

Surface, the bounding or limiting parts of a body. In the article Curve the mathematical question is treated from an historical point of view, for the purpose of showing how the leading ideas of the theory were successively arrived at. These leading ideas apply to surfaces, but the ideas peculiar to surfaces are scarcely of the like fundamental nature, being rather developments of the former set in their application to a more advanced portion of geometry; there is consequently less occasion for the historical mode of treatment. Curves in space are considered in the same article, and they will not be discussed here; but it is proper to refer to them in connexion with the other notions of solid geometry. In plane geometry the elementary figures are the point and the line; and we then have the curve, which may be regarded as a singly infinite system of points, and also as a singly infinite system of lines. In solid geometry the elementary figures are the point, the line and the plane; we have, moreover, first, that which under one aspect is the curve and under another aspect the developable (or torse), and which may be regarded as a singly infinite system of points, of lines or of planes; and secondly, the surface, which may be regarded as a doubly infinite system of points or of planes, and also as a special triply infinite system of lines. (The tangent lines of a surface are a special complex.) As distinct particular cases of the first figure we have the plane curve and the cone, and as a particular case of the second figure the ruled surface, regulus or singly infinite system of lines; we have, besides, the congruence or doubly infinite system of lines and the complex or triply infinite system of lines. And thus crowds of theories arise which have hardly any analogues in plane geometry; the relation of a curve to the various surfaces which can be drawn through it, and that of a surface to the various curves which can be drawn upon it, are different in kind from those which in plane geometry most nearly correspond to them - the relation of a system of points to the different curves through them and that of a curve to the systems of points upon it. In particular, there is nothing in plane geometry to correspond to the theory of the curves of curvature of a surface. Again, to the single theorem of plane geometry, that a line is the shortest distance between two points, there correspond in solid geometry two extensive and difficult theories - that of the geodesic lines on a surface and that of the minimal surface, or surface of minimum area, for a given boundary. And it would be easy to say more in illustration of the great extent and complexity of the subject.

In Part I. the subject will be treated by the ordinary methods of analytical geometry; Part II. will consider the Gaussian treatment by differentials, or the E, F, G analysis.

Part I.

Surfaces in General; Torses, &c.  1\. A surface may be regarded as the locus of a doubly infinite system of points - that is, the locus of the system of points determined by a single equation U = (* x, y, z, 1)"`, =0, between the cartesian co-ordinates (to fix the ideas, say rectangular co-ordinates) x, y, z; or, if we please, by a single homogeneous relation U= (* x, y, z, w)", = o, between the quadriplanar co-ordinates x, y, z, w.  The degree n of the equation is the order of the surface; and this definition of the order agrees with the geometrical one, that the order of the surface is equal to the number of the intersections of the surface by an arbitrary line. Starting from the foregoing point definition of the surface, we might develop the notions of the tangent line and the tangent plane; but it will be more convenient to consider the surface ab initio from the more general point of view in its relation to the point, the line and the plane.

2\. Mention has been made of the plane curve and the cone; it is proper to recall that the order of a plane curve is equal to the number of its intersections by an arbitrary line (in the plane of the curve), and that its class is equal to the number of tangents to the curve which pass through an arbitrary point (in the plane of the curve). The cone is a figure correlative to the plane curve: corresponding to the plane of the curve we have the vertex of the cone, to its tangents the generating lines of the cone, and to its points the tangent planes of the cone. But from a different point of view we may consider the generating lines of the cone as corresponding to the points of the curve and its tangent planes as corresponding to the tangents of the curve. From this point of view we define the order of the cone as equal to the number of its intersections (generating lines) by an arbitrary plane through the vertex, and its class as equal to the number of the tangent planes which pass through an arbitrary line through the vertex. And in the same way that a plane curve has singularities (singular points and singular tangents) so a cone has singularities (singular generating lines and singular tangent planes).

3\. Consider now a surface in connexion with an arbitrary line. The line meets the surface in a certain number of points, and, as already mentioned, the order of the surface is equal to the number of these intersections. We have through the line a certain number of tangent planes of the surface, and the class of the surface is equal to the number of these tangent planes.

But, further, through the line imagine a plane; this meets the surface in a curve the order of which is equal (as is at once seen) to the order of the surface. Again, on the line imagine a point; this is the vertex of a cone circumscribing the surface, and the class of this cone is equal (as is at once seen) to the class of the surface. The tangent lines of the surface which lie in the plane are nothing else than the tangents of the plane section, and thus form a singly infinite series of lines; similarly, the tangent lines of the surface which pass through the point are nothing else than the generating lines of the circumscribed cone, and thus form a singly infinite series of lines. But, if we consider those tangent lines of the surface which are at once in the plane and through the point, we see that they are finite in number; and we define the rank of a surface as equal to the number of tangent lines which lie in a given plane and pass through a given point in that plane. It at once follows that the class of the plane section and the order of the circumscribed cone are each equal to the rank of the surface, and are thus equal to each other. It may be noticed that for a general surface (*x, y, z, w) n, =o, of order n without point singularities the rank is a, =n(n - i), and the class is n', =n(n - I) 2; this implies (what is in fact the case) that the circumscribed cone has line singularities, for otherwise its class, that is the class of the surface, would be a(a - I), which is not =n(n - I)2.

4\. The notions of the tangent line and the tangent plane have been assumed as known, but they require to be further explained in reference to the original point definition of the sur face. Speaking generally, we may say that the points of the surface consecutive to a given point on it lie in a plane which is the tangent plane at the given point, and conversely the given point is the point of contact of this tangent plane, and that any line through the point of contact and in the tangent plane is a tangent line touching the surface at the point of contact. Hence we see at once that the tangent line is any line meeting the surface in two consecutive points, or - what is the same thing - a line meeting the surface in the point of contact counting as two intersections and in n-2 other points. But, from the foregoing notion of the tangent plane as a plane containing the point of contact and the consecutive points of the surface, the passage to the true definition of the tangent plane is ,not equally obvious. A plane in general meets the surface of the order n in a curve of that order without double points; but the plane may be such that the curve has a double point, and when this is so the plane is a tangent plane having the double point for its point of contact. The double point is either an acnode (isolated point), then the surface at the point in question is convex towards (that is, concave away from) the tangent plane; or else it is a crunode, and the surface at the point in question is then concavo-convex, that is, it has its two curvatures in opposite senses (see below, par. 16). Observe that in either case any line whatever in the plane and through the point meets the surface in the points in which it meets the plane curve, viz. in the point of contact, which qua double point counts as two intersections, and in n-2 other points; that is, we have the preceding definition of the tangent line.

5\. The complete enumeration and discussion of the singularities of a surface is a question of extreme difficulty which has not yet been solved.' A plane curve has point singularities and line singularities; corresponding to these we have for the surface isolated point singularities and isolated plane singularities, but there are besides continuous singularities applying to curves on or torses circumscribed to the surface, and it is among these that we have the non-special singularities which play the most important part in the theory. Thus the plane curve represented by the general equation (q x, y, z) n =o, of any given order n, has the non-special line singularities of inflexions and double tangents; corresponding to this the surface represented by the general equation (*Pc, y, z, w) n =o, of any given order n, has, not the isolated plane singularities, but the continuous singularities of the spinode curve or torse and the node-couple curve or torse. A plane may meet the surface in a curve having (r) a cusp (spinode) or (2) a pair of double points; in each case there is a singly infinite system of such singular tangent planes, and the locus of the points of contact is the curve, the envelope of the tangent planes the torse. The reciprocal singularities to these are the nodal curve and the cuspidal curve: the surface may intersect or touch itself along a curve in such wise that, cutting.the surface by an arbitrary plane, the curve of intersection has at each intersection of the plane with the curve on the surface (t) a double point (node) or (2) a cusp. Observe that these are singularities not occurring in the surface represented by the general equation (*) x, y, z, w) n =o of any order; observe further that in the case of both or either of these singularities the definition of the tangent plane must be modified. A tangent plane is a plane such that there is in the plane section a double point in addition to the nodes or cusps at the intersections with the singular lines on the surface.

6\. As regards isolated singularities, it will be sufficient to mention the point singularity of the conical point (or cnicnode) and the corresponding plane singularity of the conic of contact (or cnictrope). In the former case we have a point such that the consecutive points, instead of lying in a tangent plane, lie on a quadric cone, having the point for its vertex; in the latter case we have a plane touching the surface along a conic; that is, the complete intersection of the surface by the plane is made up of the conic taken twice and of a residual curve of the order n-4.

7\. We may, in the general theory of surfaces, consider either a surface and its reciprocal surface, the reciprocal surface being taken to be the surface enveloped by the polar planes (in regard to a given quadric surface) of the points of the original surface; or - what is better - we may consider a given surface in reference to the reciprocal relations of its order, rank, class and singularities. In either case we have a series of unaccented letters and a corresponding series of accented letters, and the relations between them are such that we may in any equation interchange the accented and the unaccented letters; in some cases an unaccented letter may be equal to the corresponding accented letter. Thus, let n, n' be as before the order and the class of the surface, but, instead of immediately defining the rank, let a be used to denote the class of the plane section and a' the order of the circumscribed cone; also let S, S' be numbers referring to the singularities. The form of the relations is a = a' (= rank of surface); a' =n (n - i) - S;  n' = n (n -  1) 2 - S; a = n' (n' - i) - S'; n = n' (n' -  1) 2 - S'.  In these last equations S, S are merely written down to denote proper corresponding combinations of the several numbers referring to the singularities collectively denoted by S, S respectively. The theory, as already mentioned, is a complex and difficult one.

8\. A torse or developable corresponds to a curve in space in the same manner as a cone corresponds to a plane curve: although capable of representation by an equation U = (* x, y, z, w)n =o, and so of coming under the foregoing point 1 In a plane curve the only singularities which need to be considered are those that present themselves in Pliicker's equations, for every higher singularity whatever is equivalent to a certain number of nodes, cusps, inflexions and double tangents. As regards a surface, no such reduction of the higher singularities has as yet been made.

definition of a surface, it is an entirely distinct geometrical conception. We may indeed, qua surface, regard it as a surface characterized by the property that each of its tangent planes touches it, not at a single point, but along a line; this is equivalent to saying that it is the envelope, not of a doubly infinite series of planes, as is a proper surface, but of a singly infinite system of planes. But it is perhaps easier to regard it as the locus of a singly infinite system of lines, each line meeting the consecutive line, or, what is the same thing, the lines being tangent lines of a curve in space. The tangent plane is then the plane through two consecutive lines, or, what is the same thing, an osculating plane of the curve, whence also the tangent plane intersects the surface in the generating line counting twice, and in a residual curve of the order 11-2. The curve is said to be the edge of regression of the developable, and it is a cuspidal curve thereof; that is to say, any plane section of the developable has at each point of intersection with the edge of regression a cusp. A sheet of paper bent in any manner without crumpling gives a developable; but we cannot with a single sheet of paper properly exhibit the form in the neighbourhood of the edge of regression: we need two sheets connected along a plane curve, which, when the paper is bent, becomes the edge of regression and appears as a cuspidal curve on the surface.

It may be mentioned that the condition which must be satisfied in order that the equation U=o shall represent a developable is H(U) =o; that is, the Hessian or functional determinant formed with the second differential coefficients of U must vanish in virtue of the equation U=o, or - what is the same thing - H(U) must contain U as a factor. If in cartesian co-ordinates the equation is taken in the form z - f (x, y)=o, then the condition is rt - s2=o  identically, where r, s, t denote as usual the second differential coefficients of z in regard to x, y respectively.

9\. A regulus or ruled surface is the locus of a singly infinite system of lines, where the consecutive lines do not intersect; this is a true surface, for there is a doubly infinite series of Regulus or tangent planes - in fact any plane through any one of the lines is a tangent plane of the surface, touching Surface. it at a point on the line, and in such wise that, as the tangent plane turns about the line, the point of contact moves along the line. The complete intersection of the surface by the tangent plane is made up of the line counting once and of a residual curve of the order n - 1. A quadric surface is a regulus in a twofold manner, for there are on the surface two systems of lines each of which is a regulus. A cubic surface may be a regulus (see below, par. ii).

Surfaces of the Orders 2, 3 and 4.  so. A surface of the second order or a quadric surface is a surface such that every line meets it in two points, or - what comes to the same thing - such that every plane section thereof Quadric is a conic or quadric curve. Such surfaces have been Surfaces. studied from every point of view. The only singular forms are when there is (I) a conical point (cnicnode), when the surface is a cone of the second order or quadricone; (2) a conic of contact (cnictrope), when the surface is this conic; from a different point of view it is a " surface aplatie " or flattened surface. Excluding these degenerate forms, the surface is of the order, rank and class each = 2, and it has no singularities. Distinguishing the forms according to reality, we have the ellipsoid, the hyperboloid of two sheets, the hyperboloid of one sheet, the elliptic paraboloid and the hyperbolic paraboloid (see Geometry: § Analytical). A particular case of the ellipsoid is the sphere; in abstract geometry this is a quadric surface passing through a given quadric curve, the circle at infinity. The tangent plane of a quadric surface meets it in a quadric curve having a node, that is, in a pair of lines; hence there are on the surface two singly infinite sets of lines. Two lines of the same set do not meet, but each line of the one set meets each line of the other set; the surface is thus a regulus in a twofold manner. The lines are real for the hyperboloid of one sheet and for the hyperbolic paraboloid; for the other forms of surface they are imaginary.

is. We have next the surface of the third order or cubic surface, which has also been very completely studied. Such a surface Cubic may have isolated point singularities (cnicnodes or Surfaces. points of higher singularity), or it may have a nodal line; we have thus 21 +2, =23 cases. In the general case of a surface without any singularities, the order, rank and class are=3, 6, 12 respectively. The surface has upon it 27 lines, lying by threes in 45 planes, which are triple tangent planes. Observe that the tangent plane is a plane meeting the surface in a curve having a node. For a surface of any given order n there will be a certain number of planes each meeting the surface in a curve with 3 nodes, that is, triple tangent planes; and, in the particular case where n =3, the cubic curve with 3 nodes is of course a set of 3 lines; it is found that the number of triple tangent planes is, as just mentioned, =45. This would give 135 lines, but through each line we have 5 such planes, and the number of lines is thus =27. The theory of the 27 lines is an extensive and interesting one; in particular, it may be noticed that we can, in thirty-six ways, select a system of 6X6 lines, or " double sixer," such that no two lines of the same set intersect each other, but that each line of the one set intersects each line of the other set.

A cubic surface having a nodal line is a ruled surface or regulus; in fact any plane through the nodal line meets the surface in this line counting twice and in a residual line, and there is thus on the surface a singly infinite set of lines. There are two forms.

12\. As regards quartic surfaces, only particular forms have been much studied. A quartic surface can have at most 16 conical points (cnicnodes); an instance of .such a surface is artic Fresnel's wave surface, which has 4 real cnicnodes in one of the principal planes, 4X2 imaginary ones in the other two principal planes, and 4 imaginary ones at infinity - in all 16 cnicnodes; the same surface has also 4 real +12 imaginary planes each touching the surface along a circle (cnictropes) - in all 16 cnictropes. It was easy by a mere homographic transforma tion to pass to the more general surface called the tetrahedroid; but this was itself only a particular form of the general surface with 16 cnicnodes and 16 cnictropes first studied by Kummer. Quartic surfaces with a smaller number of cnicnodes have also been considered.

Another very important form is the quartic surface having a nodal conic; the nodal conic may be the circle at infinity, and we have then the so-called anallagmatic surface, otherwise the cyclide (which includes the particular form called Dupin's cyclide). These correspond to the bicircular quartic curve of plane geometry. Other forms of quartic surface might be referred to.

Congruences and Complexes.  13\. A congruence is a doubly infinite system of lines. A line depends on four parameters and can therefore be determined so as to satisfy four conditions; if only two conditions are imposed on the line we have a doubly infinite system Congru.  of lines or a congruence. For instance, the lines meeting each of two given lines form a congruence. It is hardly necessary to remark that, imposing on the line one more condition, we have a ruled surface or regulus; thus we can in an infinity of ways separate the congruence into a singly infinite system of reguli or of torses (see below, par. 16).

Considering in connexion with the congruence two arbitrary lines, there will be in the congruence a determinate number of lines which meet each of these two lines; and the number of lines thus meeting the two lines is said to be the order-class of the congruence. If the two arbitrary lines are taken to intersect each other, the congruence lines which meet each of the two lines separate themselves into two sets - those which lie in the plane of the two lines and those which pass through their intersection. There will be in the former set a determinate number of congruence lines which is the order of the congruence, and in the latter set a determinate number of congruence lines which is the class of the congruence. In other words, the order of the congruence is equal to the number of congruence lines lying in an arbitrary plane, and its class to the number of congruence lines passing through an arbitrary point.

The following systems of lines form each of them a congruence: (A) lines meeting each of two given curves; (B) lines meeting a given curve twice; (C) lines meeting a given curve and touching a given surface; (D) lines touching each of two given surfaces; (E) lines touching a given surface twice, or, say, the bitangents of a given surface.

The last case is the most general one; and conversely for a given congruence there will be in general a surface having the congruence lines for bitangents. This surface is said to be the focal surface  of the congruence; the general surface with 16 cnicnodes first presented itself in this manner as the focal surface of a congruence. But the focal surface may degenerate into the forms belonging to the other cases A, B, C, D.

14\. A complex is a triply infinite system of lines - for instance, the tangent lines of a surface. Considering an arbitrary point in connexion with the complex, the complex lines which pass through the point form a cone; considering a plane in connexion with it, the complex lines which lie in the plane envelop a curve. It is easy to see that the class of the curve is equal to the order of the cone; in fact each of these numbers is equal to the number of complex lines which lie in an arbitrary plane and pass through an arbitrary point of that plane; and we then say order  of complex = order of curve; rank of complex=class of curve =order of cone; class of complex=class of cone. It is to be observed that, while for a congruence there is in general a surface having the congruence lines for bitangents, for a complex there is not in general any surface having the complex lines for tangents; the tangent lines of a surface are thus only a special form of complex. The theory of complexes first presented itself in the researches of Malus on systems of rays of light in connexion with double refraction.

15\. The analytical theory as well of congruences as of complexes is most easily carried out by means of the six co-ordinates of a line; viz. there are co-ordinates (a, b, c, f, g, h) connected by the equation of +bg+ch =o, and therefore such that the ratios a: b: c: f: g: h  constitute a system of four arbitrary parameters. We have thus a congruence of the order n represented by a single homogeneous equation of that order (*P, b, c, f, g, h) Th = o between the six co-ordinates; two such relations determine a congruence. But we have in regard to congruences the same difficulty as that which presents itself in regard to curves in space: it is not every congruence which can be represented completely and precisely by two  such equations (see Geometry: § Line).  The linear equation (*Vi, b, c, f, g, h) =o represents a congruence of the first order or linear congruence; such congruences are interesting both in geometry and in connexion with the theory of forces acting on a rigid body.

Curves of Curvature; Asymptotic Lines.  16\. The normals of a surface form a congruence. In any congruence the lines consecutive to a given congruence line do not in general meet this line; but there is a determinate number of consecutive lines which do meet it; or, attending for the moment to only one of these, say the congruence line is met by a consecutive congruence-line. In particular, each normal is met by a consecutive normal; this again is met by a consecutive normal, and so on. That is, we have a singly infinite system of normals each meeting the consecutive normal, and so forming a torse; starting from different normals successively, we obtain a singly infinite system of such torses. But each normal is in fact met by two consecutive normals, and, using in the construction first the one and then the other of these, we obtain two singly infinite systems of torses each intersecting the given surface at right angles. In other words, if in place of the normal we consider the point on the surface, we obtain on the surface two singly infinite systems of curves such that for any curve of either system the normals at consecutive points intersect each other; moreover, for each normal the torses of the two systems intersect each other at right angles; and therefore for each point of the surface the curves of the two systems intersect each other at right angles. The two systems of curves are said to be the curves of curvature of the surface.

The normal is met by the two consecutive normals in two points which are the centres of curvature for the point on the surface; these lie either on the same side of the point or on opposite sides, and the surface has at the point in question like curvatures or opposite curvatures in the two cases respectively (see above, par. 4).

17\. In immediate connexion with the curves of curvature we have the so-called asymptotic curves (Haupt-tangentenlinien). The tangent plane at a point of the surface cuts the surface in a curve having at that point a node. Thus we have at the point of the surface two directions of passage to a consecutive point, or, say, two elements of arc; and, passing along one of these to the consecutive point, and thence to a consecutive point, and so on, we obtain on the surface a curve. Starting successively from different points of the surface we thus obtain a singly infinite system of curves; or, using first one and then the other of the two directions, we obtain two singly infinite systems of curves, which are the curves above referred to. The two curves at any point are equally inclined to the two curves of curvature at that point, or - what is the same thing - the supplementary angles formed by the two asymptotic lines are bisected by the two curves of curvature. In the case of a quadric surface the asymptotic curves are the two systems of lines on the surface.

Geodetic Lines.  18\. A geodetic line (or curve) is a shortest curve on a surface; more accurately, the element of arc between two consecutive points of a geodetic line is a shortest arc on the surface. We are thus led to the fundamental property that at each point of the curve the osculating plane of the curve passes through the normal of the surface; in other words, any two consecutive arcs PP', P'P"  are in piano with the normal at P'. Starting from a given point P on the surface, we have a singly infinite system of geodetics proceeding along the surface in the direction of the several tangent lines at the point P; and, if the direction PP' is given, the property gives a construction by successive elements of arc for the required geodetic line.

Considering the geodetic lines which proceed from a given point P of the surface, any particular geodetic line is or is not again intersected by the consecutive generating line; if it is thus intersected, the generating line is a shortest line on the surface up to, but not beyond, the point at which it is first intersected by the consecutive generating line; if it is not intersected, it continues a shortest line for the whole course.

In the analytical theory both of geodetic lines and of the curves of curvature, and in other parts of the theory of surfaces, it is very convenient to consider the rectangular co-ordinates x, y, z of a point of the surface as given functions of two independent parameters p, q; the form of these functions of course determines the surface, since by the elimination of p, q from the three equations we obtain the equation in the co-ordinates x, y, z. We have for the geodetic lines a differential equation of the second order between p and q;  the general solution contains two arbitrary constants, and is thus capable of representing the geodetic line which can be drawn from a given point in a given direction on the surface. In the case of a quadric surface the solution involves hyperelliptic integrals of the first kind, depending on the square root of a sextic function.

Curvilinear Co-ordinates.  19\. The expressions of the co-ordinates x, y, z in terms of p, q  may contain a parameter r, and, if this is regarded as a given constant, these expressions will as before refer to a point on a given surface. But, if p, q, r are regarded as three independent parameters x, y, z will be the co-ordinates of a point in space, determined by means of the three parameters p, q, r; these parameters are said to be the curvilinear co-ordinates, or (in a generalized sense of the term) simply the co-ordinates of the point. We arrive otherwise at the notion by taking p, q, r each as a given function of x, y, z; say we have p=fi(x, y, z), q=f 2 (x, y, z), r =f 3 (x, y, z), which equations of course lead to expressions for p, q, r each as a function of x, y, z. The first equation determines a singly infinite set of surfaces: for any given value of p we have a surface; and similarly the second and third equations determine each a singly infinite set of surfaces. If, to fix the ideas, fi, f2, fa are taken to denote each a rational and integral function of x, y, z, then two surfaces of the same set will not intersect each other, and through a given point of space there will pass one surface of each set; that is, the point will be determined as a point of intersection of three surfaces belonging to the three sets respectively; moreover, the whole of space will be divided by the three sets of surfaces into a triply infinite system of elements, each of them being a parallelepiped.

Orthotomic Surfaces; Parallel Surfaces.  20\. The three sets of surfaces may be such that the three surfaces through any point of space whatever intersect each other at right angles; and they are in this case said to be orthotomic. The term curvilinear co-ordinates was almost appropriated by Lame, to whom this theory is chiefly due, to the case in question: assuming that the equations p=fi(x, y, z), q=f 2 (x, y, z), r=f 3 (x, y, z) refer to a system of orthotomic surfaces, we have in the restricted sense p, q, r as the curvilinear co-ordinates of the point.

An interesting special case is that of confocal quadric surfaces. The general equation of a surface confocal with the ellipsoid x 2 y2 z2 _ x2 y z2 a2 + b 2 + c2 = I is a2 +0 + b2 +0 \+ c2 +0 -  I; and, if in this equation we consider x, y, z as given, we have for 0 a cubic equation with three real roots p, q, r, and thus we have through the point three real surfaces, one an ellipsoid, one a hyperboloid of one sheet, and one a hyperboloid of two sheets.

21\. The theory is connected with that of curves of curvature by Dupin's theorem. Thus in any system of orthotomic surfaces each surface of any one of the three sets is intersected by the surfaces of the other two sets in its curves of curvature.

22\. No one of the three sets of surfaces is altogether arbitrary: in the equation p=f i (x, y, z), p is not an arbitrary function of x, y, z, but it must satisfy a certain partial differential equation of the third order. Assuming that p has this value, we have q=f 2 (x, y, z) and r=f 3 (x, y, z) determinate functions of x,y,z such that the three sets of surfaces form an orthotomic system.

23\. Starting from a given surface, it has been seen (par. 16) that the normals along the curves of curvature form two systems of torses intersecting each other, and also the given surface, at right angles. But there are, intersecting the two systems of tomes at right angles, not only the given surface, but a singly infinite system of surfaces. If at each point of the given surface we measure off along the normal one and the same distance at pleasure, then the locus of the points thus obtained is a surface cutting all the normals of the given surface at right angles, or, in other words, having the same normals as the given surface; and it is therefore a parallel surface to the given surface. Hence the singly infinite system of parallel surfaces and the two singly infinite systems of torses form together a set of orthotomic surfaces.

The Minimal Surface.  24\. This is the surface of minimum area - more accurately, a surface such that, for any indefinitely small closed curve which can be drawn on it round any point, the area of the surface is less than it is for any other surface whatever through the closed curve. It at once follows that the surface at every point is concavo-convex; for, if at any point this was not the case, we could, by cutting the surface by a plane, describe round the point an indefinitely small closed plane curve, and the plane area within the closed curve would then be less than the area of the element of surface within the same curve. The condition leads to a partial differential equation of the second order for the determination of the minimal surface: considering z as a function of x, y, and writing as usual p, q, r, s, t for the first and the second differential coefficients of z in regard to x, y respectively, the equation (as first shown by Lagrange) is (1 + q 2)r - 2pgs + (I + p 2)t = 0, or, as this may also be written, y/ +p2+q2 + d p 0\. The general integral contains of course arbitrary functions, and, if we imagine these so determined that the surface may pass through a given closed curve, and if, moreover, there is but one minimal surface passing through that curve, we have the solution of the problem of finding the surface of minimum area within the same curve. The surface continued beyond the closed curve is a minimal surface, but it is not of necessity or in general a surface of minimum area for an arbitrary bounding curve not wholly included within the given closed curve. It is hardly necessary to remark that the plane is a minimal surface, and that, if the given closed curve is a plane curve, the plane is the proper solution; that is, the plane area within the given closed curve is less than the area for any other surface through the same curve. The given closed curve is not dx +pz +q2  of necessity a single curve: it may be, for instance, a skew polygon of four or more sides.

The partial differential equation was dealt with in a very remarkable manner by Riemann. From the second form given qdx - pdy  above it appears that we have p2 q2 = a complete diffe rential, or, putting this = dl', we introduce into the solution a variable 1, which combines with z in the forms z it (i = - I). The boundary conditions have to be satisfied by the determination of the conjugate variables n, n' as functions of z+ i?, z - it , or, say, of Z, Z respectively, and by writing S, S to denote x+iy, x - iy respectively. Riemann obtains finally two ordinary differential equations of the first order in S, S', n, ,', Z, Z', and the results are completely worked out in some very interesting special cases.

(A. CA.) Part Ii.

We proceed to treat the differential geometry of surfaces, a study founded on the consideration of the expression of the lineal element in terms of two parameters, u, v, ds 2 = Edu 2 + 2Fdudv+ Gdv2, u= const, v = const, being thus systems of curves traced on the surface. This method, which may be said to have been inaugurated by Gauss in his classical paper published in 1828, Disquisitiones generales circa superficies curves, has the great advantage of dealing in the most natural way with all questions connected with geodetics, geodetic curvature, geodetic circles, &c. - in fact, all relations of lines on a surface which can be formulated without reference to anything external to the surface. All such relations when deduced for any particular surface can be at once generalized in their application, holding good for any other surface which has the same expression for its lineal element; e.g. relations involving great circles and small circles on a sphere furnish us with corresponding relations for geodetics and geodetic circles on any synclastic surface of constant specific curvature.

t. Gauss begins by introducing the conception of the integral curvature (curvatura Integra) of any portion of a surface. This he defines to be the area of the corresponding portion of a sphere of unit radius, traced out by a radius drawn parallel to the normal at each point of the surface; i.e. it is ff ds/RR' where R, R' are the principal radii of curvature. The quotient obtained by dividing the integral curvature of a small portion of the surface round a point by the area of that portion, that is 1 /RR', he naturally calls the measure of curvature or the specific curvature at the point in question. He proceeds to establish his leading proposition, that this specific curvature at any point is expressible in terms of the E, F and G which enter into the equation for the lineal element, together with their differential coefficients with respect to the variables, u and v.  It is desirable to make clear the exact significance of this theorem. Of course, for any particular surface, the curvature can be expressed in an indefinite variety of ways. The speciality of the Gaussian expression is that it is deduced in such a manner as to hold good for all surfaces which have the same expression for the lineal element. The expression for the specific curvature, which is in general somewhat elaborate, assumes a very simple form when a system of geodetics and the system of their orthogonal trajectories are chosen for the parameter curves, the parameter u being made the length of the arc of the geodetic, measured from the curve, u = o selected as the standard. If this be done the equation for the lineal element becomes ds 2 = du e + P 2 dv 2 , and that for the specific curvature (RR 1)- 1 = - P -1 d 2 Pldu 2. By means of this last expression Gauss then proves that the integral curvature of a triangle formed by three geodetics on the surface can be expressed in terms of its angles, and is equal to A+ B+ C - 7r.

This theorem may be more generally stated The 'integral curvature of any portion of a surface=27r - Zdi round the contour of this portion, where di denotes the angle of geodetic contingence of the boundary curve. The angle of geodetic contingence of a curve traced on a surface may be defined as the angle of intersection of two geodetic tangents drawn at the extremities of an element of arc, an angle which may be easily proved to be the same as the projection on the tangent plane of the ordinary angle of contingence. The geodetic curvature, p-', is thus equal to the ordinary curvature multiplied by cos 4), 4 being the angle the osculating plane of the curve makes with the tangent plane.

'Gauss's theorem may be established geometrically in the following simple manner: If we draw successive tangent planes along the curve, these will intersect in a system of lines, termed the conjugate  tangents, forming a developable surface. If we unroll this developable then di = do - d, where di is the angle of geodetic con tingence, do the angle between two consecutive conjugate tangents, y the angle the conjugate tangent makes with the curve. Therefore, as 4) returns to its original value when we integrate round the curve, we have di= Edo. This equation holds for both the curve on the given surface and the representative curve on the sphere. But the tangent planes along these curves being always parallel, their successive intersections are so also; therefore Zdo is the same for both; consequently Zdi for the curve on the surface =Edi for the representative curve on the sphere. Hence integral curvature of curve of surface = area of representative curve on sphere, = 27r - Zdi on sphere by spherical geometry, = 27r - Zdi for curve on surface.

A useful expression for the geodetic curvature of one of the curves, v= const, can be obtained. If a curve receive a small displacement on any surface, so that the displacements of its two extremities are normal to the curve, it follows, from the calculus of variations, that the variation of the length of the curve =fp1 Snds where p-1 is the geodetic curvature, and Sn the normal component of the displacement at each point. Applying this formula to one of the v curves, we find S f Pdv = f (dP/du)Sudv = S length of curve = f p1 6uPdv, and as Su is the same for all points of the curve, p1 = P'1dP/du. We can deduce immediately from this expression Gauss's value for the specific curvature. For applying his theorem to the quadrilateral formed by the curves u, u l, v, v , and remembering that Zdi along a geodetic vanishes, we have ff (RR')- 1 Pdudv = - Zdi for curve BC - Zdi for curve DA, = - Zppi ds for curve BC+ Zpi ds for curve AD, - J du Pdv for curve BC+ f d uPdv for curve AD, P J du)ul - duJu dv, therefore passing to the limit P/RR' = - d2P/du2.

Gauss then proceeds to consider what the result will be if a surface be deformed in such a way that no lineal element is altered. It is easily seen that this involves that the angle at which two curves on the surface intersect is unaltered by this deformation; and since obviously geodetics remain geodetics, the angle of geodetic contingence and consequently the geodetic curvature are also unaltered. It therefore follows from his theorem that the integral curvature of any portion of a surface and the specific curvature at any point are unaltered by non-extensional deformation.

Geodetics and Geodetic Circles.  A geodetic and its fundamental properties are stated in part I., where it is also explained in that article within what range a geodetic possesses the property of being the shortest path between two of its points. The determination of the geodetics on a given surface depends upon the solution of a differential equation of the second order. The first integral of this equation, when it can be found for any given class of surfaces, gives us the characteristic property of the geodetics on such surfaces. The following are some of the wellknown classes for which this integral has been obtained: (1) quadrics; (2) developable surfaces; (3) surfaces of revolution.

1\. Quadrics. - Several mathematicians about the middle of the 19th century made a special study of the geometry of the lines of curvature and the geodetics on quadrics, and were rewarded by the discovery of many wonderfully simple and elegant analogies between their properties and those of a system of confocal conics and their tangents in plano. As explained above, the lines of curvature on a quadric are the systems of orthogonal curves formed by its intersection with the two systems of confocal quadrics. Joachimsthal showed that the interpretation of the first integral of the equation for geodetics on a central quadric is, that along a geodetic pD =constant (C,) p denoting the perpendicular let fall from the centre on the tangent plane, and D the semidiameter drawn parallel to the element of the geodetic, the envelope of all geodetics having the same C being a line of curvature. In particular, all geodetics passing through one of the real umbilics (the four points where the indicatrix is a circle) have the same C.

Michael Roberts pointed out that it is an immediate consequence of the equation pD = C, that if two umbilics, A and B (selecting two not diametrically opposite), be joined by geodetics to any point P on a given line of curvature, they make equal angles with such line of curvature, and consequently that, as P moves along a line of curvature, either PA+PB or PA - PB remains constant. Or, conversely, that the locus of a point P on the surface, for which the sum or difference of the geodetic distances PA and PB is constant, is a line of curvature. It follows that if the ends of a string be fastened at the two umbilics of a central quadric, and a style move over the surface keeping the string always stretched, it will describe a line of curvature.

Another striking analogue is the following: As, in plano, if a variable point or an ellipse be joined to the two foci S and H, tan a PSH tan 2PHS=const, and for the hyperbola tan 2PSH/tan 2PHS=const, so for a line of curvature on a central quadric, if P be joined to two umbilics S and H by geodetics, either the product or the ratio of the tangents of PSH and 2PHS will be constant.

Chasles proved that if an ellipse be intersected in the point A by a confocal hyperbola, and from any point P on the hyperbola tangents PT, PT' be drawn to the ellipse, then the difference of the arcs of the ellipse TA, T'A = the difference of the tangents PT, PT'; and subsequently Graves showed that if from any point P on the outer of two confocal ellipses tangents be drawn to the inner, then the excess of the sum of the tangents PT, PT' over the intercepted arc TT' is constant. Precisely the same theorems hold for a quadric replacing the confocals by lines of curvature and the rectilineal tangents by geodetic tangents. Hart still further developed the analogies with confocal conics, and established the following: If a geodetic polygon circumscribe a line of curvature, and all its vertices but one move on lines of curvature, this vertex will also describe a line of curvature, and when the lines of curvature all belong to the same system the perimeter of the polygon will be constant.

## 2\. Geodetics on Developable Surfaces

On these the geodetics are the curves which become right lines when the surface is unrolled into a plane. From this property a first integral can be immediately deduced.

## 3\. Geodetics on Surfaces of Revolution

In all such the geodetics are the curves given by the equation r sin 4)4) =const, r being the perpendicular on the axis of revolution, 4) the angle at which the curve crosses the meridian.

The general problem of the determination of geodetics on any surface may be advantageously treated in connexion with that of " parallel " curves. By " parallel " curves are meant curves whose geodetic distances from one another are constant - in other words, the orthogonal trajectories of a system of geodetics. In applying this method the determination of a system of parallel curves comes first, and the determination of the geodetics to which they are orthogonal follows as a deduction. If (u, v) =const be a system of parallel curves, it is shown that 4) must satisfy the partial differential equation E (2_2F () G (') 2=EGI_ F.

If (u, v, a) =const be a system of parallel curves satisfying this equation, then d4/da=const is proved to represent the orthogonal geodetics. The same method enables us to establish a result first arrived at by Jacobi, that whenever a first integral of the differential equation for geodetics can be found, the final integral is always reducible to quadratures. In this method r i s corresponds to the characteristic function in the Hamiltonian dynamics, the geodetics being the paths of a particle confined to the surface when no extraneous forces are in action.

The expression for the lineal element on a quadric in elliptic co-ordinates suggested to Lionville the consideration of the class of surfaces for which this equation takes the more general form ds 2 = (U - V)(U 1 2 du 2 +V 1 2 dv 2), where U, U 1 are functions of u, and V, V 1 functions of v, and shows that, for this class, the first integral of the equation of the parallels is immediately obtainable, and hence that of the corresponding geodetics. It is to be remarked that for this more general class of surfaces the theorems of Chasles and Graves given above will also hold good.

Geodetics on a surface corresponding to right lines on a plane, the question arises what curves on a surface should be considered to correspond to plane circles. There are two claimants for the position: first, the curves described by a point whose geodetic distance from a given point is constant; and, second, the curves of constant geodetic curvature.

On certain surfaces the curves which satisfy one of these conditions also satisfy the other, but in general the two curves must be carefully distinguished. The property involved in the second definition is more intrinsic, and we shall therefore, following Lionville, call the curves pos- X sessing it geodetic circles. It may be noted that geodetic circles, except on surfaces of constant specific curvature, do not return back upon themselves like circles in Plano.  As a particular instance, a geodetic on an ellipsoid (which is, of course, a geodetic circle of zero curvature), starting from an umbilic, when it returns again, as it does to that umbilic, makes a finite angle with its original starting position. As to the curve described by a point whose geodetic distance from a given centre is constant, Gauss showed from the fundamental property of a geodetic that this curve resembles the plane circle in being everywhere perpendicular to its radius. In the same way it holds that the curve described by a point the sum (or difference) of whose geodetic distances fromtwo given points (foci) is constant, resembles the plane ellipse (or hyperbola) in the property that it bisects at every point the external (or internal) angle between the geodetic focal radii, and, as a consequence, that the curves on any surface answering to confocal ellipses and hyperbolas intersect at right angles. The equation for the lineal element enables us to dis-, cuss geodetic circles on sur-; faces of constant specific curvature; for we have seen that if we choose as parameters geodetics and their orthogonal trajectories, the equation becomes ds 2 = du e + P 2 du 2; and since (RR')- 1 = - P-1d2P/due, and here (RR')- 1 = =a-2, it follows P =A cos ua 1+ B sin ua 1, or P = A cosh ua 1 -}-B sinh ua 1 , according as the surface is synclastic or anticlastic. If a geodetic circle (curvature k' 1 ) be chosen for the starting curve u = o, and if v be made the length of the arc OY, intercepted on this circle by the curve v = const (see fig. I), then A and B can be proved to be independent of u and P = cos ua 1 -{-ak1 sin ua° 1 for a synclastic surface, P = cosh ua 1 sinh ua 1 for an anticlastic sur face. It follows from the expression for the geodetic curvature p-1= P1 dP/du that in both classes of surfaces all the other orthogonal curves u = const will be geodetic circles. It also appears that on a synclastic surface of constant specific curvature all the geodetics normal to a geodetic circle converge to a point on either side as on a sphere, and can be described with a stretched string taking either of these points as centre, the length of the string being a tan1 ale1 (see fig. 2). These normals will be all cut orthogonally by an equator, that is, by a geodetic circle of zero curvature.

For anticlastic surfaces, however, we must distinguish two cases. If the curvature k1 of the geodetic circle >a 1 the geodetic normals meet in a point on the concave side of the geodetic circle, and can be described as on the synclastic by a stretched string, the length of the string being a tanh1 ak -1, but in this case the geodetic normals have no equator (see fig. 3). If on the other hand the curvature of the geodetic circle be <a1 the normals do not meet on either side, but do possess an equator, and at this equaFIG. 4.

tor the geodetic normals come nearer together than they do anywhere else (see fig. 4).

On a synclastic surface of constant specific curvature a l two near geodetics proceeding from a point always meet again at the geodetic FIG. I.

FIG. 2.

FIG. 3.

distance Ira; and more generally for any synclastic surface whose specific curvature at every point lies between the limits a2 and b-2  two near geodetics proceeding from a point always meet again at a geodetic distance intermediate in value between ira and irb. On an anticlastic surface two near geodetics proceeding from a point never meet again.

Representation of Figures on a Surface by Corresponding Figures on a Plane; Theory of Maps.  The most valuable methods of effecting such representation are those in which small figures are identical in shape with the figures which they represent. This property is known to belong to the representation of a spherical surface by Mercator's method as well as to the representation by stereographic projection. The problem of effecting this conformable representation is easily seen to be equivalent to that of throwing the expression for the lineal element into what is known in the theory of heat conduction as the isothermal form ds 2 =X(du 2 -4-dv 2), for we have then only to choose for the representative point on the plane that whose rectangular co-ordinates are x = u, y = v. A curious investigation has been made by Beltrami - when is it possible to represent a surface on a plane in such a way that the geodetics on the surface shall correspond to the right lines on the plane (as, for example, holds true when a spherical surface is projected on a plane by lines through its centre)? He has proved that the only class of surface for which such representation is possible is the class of uniform specific curvature.

Just as the intrinsic properties of a synclastic surface of uniform specific curvature are reducible to those of a particular surface of this type, i.e. the sphere, so we can deal with an anticlastic surface of constant specific curvature, and reduce its properties to a particular anticlastic surface. A convenient surface to study for this purpose is that known as the pseudosphere, formed by the revolution of the tractrix (an involute of the catenary) round its base (see fig. 5).

0 z Its equations are r =a sin 0, axis of revolution z=a(cos 4+log tan 1 2 -4). This FIG. 5. surface can be conformably repre sented as a plane map by choosing x' =6.) where w is the longitude of the point and y'=a/sin 0. It will then be found that ds=ads'/y', where ds =lineal element on the surface, ds'=same on the map. It easily appears that geodetic circles on the surface are represented by circles on the map, the angle 4' at which these circles cut the base depending only upon the a .x  FIG. 6.

curvature of the geodetic circle, cos ,y being equal top 1. As a particular case it follows that the geodetics on the surface are represented by those special circles on the map whose centres lie on the base (see fig. 6). The geodetic distance between two points P and Q on the surface is represented by the logarithm of the anharmonic function AP'BQ', where P'Q' are the representing points on the map, A B the points in which the circle on the map which passes through P' and Q' and has its centre on the base cuts the base. The perimeter (1) of a geodetic circle of curvature p1 turns out to be 21rapl-1 (a 2 - p 2), and its area (1p 1 -27r)a 2. The geometry of coaxal circles in piano accordingly enables us to demonstrate anew by means of the pseudosphere the properties which we have shown to hold good in all anticlastic surfaces of constant curvature. Thus the system of geodetics cutting orthogonally a geodetic circle C will be represented on the map by circles having their centres on the base, and cutting a given circle C' orthogonally, i.e. by a coaxal system of circles. We know that the other orthogonal trajectories of this last system are another coaxal system, and therefore, going back to the pseudosphere, we learn that if a system of geodetics be drawn normal to a geodetic circle, all the orthogonals to this system are, geodetic circles. It is to be noted that while every point on the surface has its representative on the map, the converse does not hold. It is only points lying above the line y' =a which have their prototypes on the surface, the portion of the plane below this line not answering to any real part of the surface. If we take any curve C' on the map crossing this line, the part of the curve above this line has as its prototype a curve on the surface. When C' reaches this line, C reaches the circular base of the pseudosphere, and there terminates abruptly. The distinction between the two cases of a geodetic circle with curvature greater and one with curvature less than a1 also comes out clearly. For if curvature of C> a1 the map circle C' lies entirely above the base, and the coaxal system cutting C' orthogonally passes through a real point; therefore C has a centre. If curvature of C <a1 the map circle C' intersects the base, the coaxal system cutting C' orthogonally does not intersect in a real point, and C has accordingly no centre. It is of interest to examine in what way a pseudosphere differs from a plane as regards the behaviour of parallel lines. If on a plane a geodetic AB (i.e. a right line) be taken, and another geodetic constantly pass through a point P and revolve round P, it will always meet AB in the point except in the particular position. On the pseudosphere, if we carry out the corresponding construction, the position of the non-intersecting geodetic is not unique, but all geodetics drawn within a certain angle fail to meet the geodetic AB.

Minimal Surfaces.  From the definition given in part I. readily follows the wellknown property of these surfaces - that the two principal curvatures are at every point of such a surface equal and opposite. For familiar instances of the class we have the surface formed by the revolution of a catenary round its base called by French mathematicians the alysseide, and the right conoid, z =a tan 1 (y/x), formed by the successive edges of the steps of a spiral staircase. Monge succeeded in expressing the co-ordinates of the most general minimal surface in two parameters, and in a form in which the variables are separated. The separation of the variables in the expression signifies that every minimal surface belongs to the class of surfaces which can be generated by a movement of translation of a curve. Enneper has thrown the expression for the co-ordinates into the following convenient forms: - x =2 (u 2)f(u)du+2 f(i - v) (v)dv, y= f r - zi J (i +u 2)f(u) du -  Z i f(i -+v2)4'(v)dv, z = fuf(u)du+ fv (v)dv.  It is noteworthy that the expression for the lineal element on a minimal surface assumes the isothermal form ds 2 =X(du 2 dv2) -  (I) when the curves u= const, v= const are so chosen as to be the lines of curvature; and (2) when they are chosen to be the lines in which the surface is intersected by a system of parallel planes and the orthogonal trajectories of these lines. It is easily proved that a minimal surface possesses the property of being conformable to its spherical representation. For since the indicatrix at every point is a rectangular hyperbola, the angle between the elements of two intersecting curves=angle between their conjugate tangents; but this = angle between conjugate tangents to representative curves on sphere=angle between these curves themselves.

The problem of finding a minimal surface to pass through a given curve in space, known as Plateau's problem, possesses an exceptional interest from the circumstance that it can be always exhibited to the eye in the following way by an actual physical experiment. Dip a wire having the form of the given curve in a soap-bubble solution, and the film adhering to the wire when it is withdrawn is the surface required. This is evident, since from the theory of surface-tension we know that a very thin film must assume that form for which the area of its surface is the least possible. The same theory also furnishes us with an elementary proof of the characteristic property that the sum of the curvatures is everywhere zero, inasmuch as the normal pressure on the film, here zero, is known to be proportional to the surface-tension multiplied by the sum of the curvatures.

Riemann, adopting a method depending upon the use of the complex variable, has succeeded in solving Plateau's problem for several interesting cases, e.g. I° when the contour consists of three infinite right lines; 2° when it consists of a gauche quadrilateral; and 3° when it consists of any two circles situated in parallel planes. (For Lie's investigations in this domain, see Theory of groups.) Non-extensional Deformation.  We have already explained what is meant by this term. It is a  subject to which much study has been devoted, connecting itself, as it does, with the work of Gauss in pure geometry on the one hand and with the theory of elasticity on the other. Several questions have been opened up: (I) What are the conditions which must be fulfilled by two surfaces such that one can be " deformed " so as to fit on the other? (2) What instances have we of known surfaces applicable to one another? (3) What surfaces are applicable to themselves? (4) In regard to infinitely small deformations, what are the differential equations which must be satisfied by the displacements? (5) Under what circumstances can a surface not be deformed? Can a closed surface ever be deformed?

I. Of course if two surfaces are applicable we must be able to get two systems of parameter curves u = const, v = const, on the first n{,ation of geo Line y' =  surface, and two systems on the second, such that the equation for the lineal element, when referred to these, may have an identical form for the two surfaces. The problem is now to select these corresponding systems. We may conveniently take for the co-ordinate u the specific curvature on each surface, and choose for v the function du/dn which denotes the rate of increase of u along a direction normal to the curve u=const. Then, since at corresponding points both u and v will be the same for one surface as for the other, if the surfaces are applicable, E, F and G, in the equation ds 2 = Edu2+2Fdudv+Gdv2, must be identical for the two surfaces. Clerk Maxwell has put the geometrical relation which exists between two applicable surfaces in the following way: If we take any two corresponding points P and P' on two such surfaces, it is always possible to draw two elements through P parallel to conjugate semi-diameters of the indicatrix at P, such that the corresponding elements through P' shall be parallel to conjugate semi-diameters of the indicatrix at P'. The curves made up of all these elements will divide the two surfaces into small parallelograms, the four parallelograms having P as common vertex being identical in size and shape with the four having P' as vertex. Maxwell regards the surfaces as made up in the limit of these small parallelograms. Now, in order to render these surfaces ready for application, the first step would be to alter the angle between two of the planes of the parallelograms at P, so as to make it equal to that between the corresponding planes at P'. If this be done it is readily seen that all the angles between the other planes at P and P', and at all other corresponding points, will become equal also. The curves which thus belong to the conjugate systems common to the two surfaces may be regarded as lines of bending.  2\. Any surface of uniform specific curvature, whether positive or negative, is applicable to another surface of the same uniform specific curvature in an infinite variety of ways. For if we arbitrarily choose two points, 0 and 0', one on each surface, and two elements, one through each point, we can apply the surfaces, making O and 0' corresponding points and the elements corresponding elements. This follows from the form of the equation of the lineal element, which is for synclastic surfaces ds 2 =du 2 +a 2 sin2(ua-1)dv2, and for anticlastic, ds 2 =du 2 +a 2 sinh 2 (ua - ')dv 2, and is therefore identical for the two surfaces in question. Again, a ruled surface may evidently be deformed by first rotating round a generator, the portion of the surface lying to one side of this generator, then round the consecutive generator, the portion of the surface lying beyond this again, and so on. It is clear that in such deformation the rectilinear generators in the old surface remain the rectilinear generators in the new; but it is interesting to note that two ruled surfaces can be constructed which shall be applicable, yet so that the generators will not correspond. For, deform a hyperboloid of one sheet in the manner described, turning the portions of the surface round the consecutive generators of one system, and then deform the hyperboloid, using the generators of the other system. The two surfaces so obtained are, of course, applicable to one another, yet so that their generators do not now correspond. Conversely Bonnet has shown that, whenever two ruled surfaces are thus applicable, without correspondence of generators, they must be both applicable to the same hyperboloid of one sheet. The alysseide is a good example of a surface of revolution applicable to a ruled surface, in this case the right circular conoid, the generators of the conoid coinciding with the meridians of the alysseide.

3\. As instances of surfaces applicable to themselves, we may take surfaces of uniform specific curvature, as obviously follows from the reasoning already given; also surfaces of revolution, inasmuch as any such surface can be turned round its axis and still fit upon its old position. Again, helicoidal surfaces possess this property. A helicoidal surface means that traced out by a rigid wire, which is given a screw motion round a fixed axis, or, which comes to the same thing, the surface made up of a system of helices starting from the points of a given curve, all having the same axis and the same interval between the successive threads. The applicability of such a surface to itself, if given a screw motion round the axis, is evident from the law of its formation.

4\. The possible small variations, n, ?' of the points of a surface when it is subject to a small inextensional deformation are conditioned by the equation dxd4-}-dydn+dzdi =0, or making x and y  the independent variables, i dx2 + dxd y (dy + d +p d y + g dz) + dy 2 (,+ gdyl ° O.

From this it follows that the three equations must separately hold dE + pd ?' -, d ? + an +p d ? + ga ? - 0 dn, + ga ? - 0 dx dy dx dy dx - dy dy - '  Accordingly, the determination of a possible small deformation cf a given surface is reduced to the analytical problem of finding three functions, n, of the variables x and y to satisfy these equations. Changing the co-ordinates to a and f where a= const, /3 = const, are the curves of inflexion on the surface, the solution of the equations can be shown to depend upon that of the equation d 2 w, /dads = Xw, where X is a function of a and depending on the form of the surface. The last equation can be integrated, and the possible deformation determined in the case of a spherical surface, or of any surface of uniform specific curvature. It is easily shown that if we have determined the displacements for any surface S we can do so for any surface obtained from S by a linear transformation of the variables. For let x'=alx=bly+c,z±d,, y=a 2x +b2y+c2 z +d 2, '=' 3 x +b3y+c3z+d3, then the displacements y = AIE+Bin+C1", n ' =A 2S+ B 2n+ C 2J, J ' = A 3 + B3n+C3J, where A B, &c., are the minors of the determinant [a l b c3], will evidently satisfy the equation dx'dE' + dy'dn' + dz'dl' = 0.

Accordingly the known solution for a sphere furnishes us with a solution for any quadric. Moutard has pointed out a curious connexion between the problem of small deformation and that of the applicability of two finitely different surfaces.

For if dxd+dydn+dz(R-=0, it follows that if k be any constant, {d(x+k)}2+{d(y -kn)}2 +{d(z+k?')}2 =1d(x - kE)}2 \+ 1d(y - kn)12+{d(z - k0}2.  Consequently, if we take two surfaces such that for the first X = Y = y +kn, Z = z+4, and for the second X' = x - k?, Y' = y - kn, Z' = then dX2 +dY 2 -j -dZ 2 = dX' 2 +dY' 2 = dZ'2, and therefore the new surfaces are applicable.

5\. Jellett and Clerk Maxwell have shown by different methods that, if a curve on a surface be held fixed, there can be no small deformation, except this curve be a curve of inflexion. This may be also proved thus: There can be no displacement of the tangent planes along the fixed curve, for, at any point of the curve the geodetic curvature cannot alter; but in present case the ordinary curvature of the curve is also fixed, therefore their ratio is constant, so that Scos0 = - sin 6SO =0, where 0 is the angle which the osculating plane makes with the tangent plane; therefore unless sin 0=0, as it is along a curve of inflexion, SO = 0, and therefore the tangent plane at each point is unaltered. Hence it can be shown that along the given curve not only, n, vanish, but also their differential coefficients of all orders, and therefore no displacement is possible. The question has been much discussed Can a closed synclastic surface be deformed? There seems to be a prevalent opinion amongst mathematicians that such deformation is always impossible, but we do not think any unimpeachable demonstration of this has yet been given. It is certain that a complete spherical surface does not admit of inextensive deformation, for if it did it would follow from Gauss's theorem that the new surface would have a uniform specific curvature. Now, it is not difficult to prove that the only closed surface possessing this property is the sphere itself, provided that the surfaces in question be such that all their tangent planes lie entirely outside them. We can then, by the method of linear transformation already given, extend the theorem of the impossibility of deformation to any ellipsoid.

The theorem that a sphere is the only closed surface of constant specific curvature may, we suggest, be established by means of the following two propositions, which hold for integration on any closed surface, p being the perpendicular from the origin on the tangent plane ff (I 'I/R'') dS = 2ff pdS /RR' (i) 2 ff dS = if p(I/R + i /R')dS. (2) Now multiply both sides of the first equation by the constant 1/ RR', and subtract the second, and we get ff {(R'/R)I - (R/R')4}'-dS+ff p(I/R1 - I/R'i)2dS=0, which is impossible unless R' = R everywhere, since in accordance with the proviso p is everywhere positive.

Theorems (r) and (2) are deduced by Jellett by means of the calculus of variations in his treatise on that subject. They may also be very simply proved thus: Draw normals to the surface along the contours of the small squares formed by lines of curvature, and let these meet successive parallel surfaces at distances dn, then the volume bounded by two parallel surfaces = ff (dS f o dn(I + n/R) (I + n/R') ff dS(n + 2n 2 (I / R \+ I/R') + an 3 /RR'); but taking origin inside, the perpendiculars let fall from 0 on a tangent plane to the outer surface = p+n on account of the parallelism of the surfaces. Also dS for outer surface =dS(I +n/R) (I +n/R'); therefore volume in question = 31f (p + n) (I + n/R) (I + nIR')dS - a ff pdS  =inf./ +R+R')dS+ 3n2ff P + I dS + 3 n3ffRR' dS.  Hence equating coefficients of the powers of n - f fp (i/R-{-i/R')dS =2 f fdS, and ff 2pdS/RR' = f f (1/R + i/R')dS.

References to the original memoirs will be found in Salmon's Analytical Geometry of Three Dimensions, Frost's Solid Geometry and, more completely, in Darboux's Lecons sur la theorie generale des surfaces. (J. Pu.; F. Pu.)