IN THE THEORY OF DIFFERENTIAL EQUATIONS. 537 



one definite surface. A curve upon a surface corresponds to a point upon a curve : by giving 

 one point, we select one curve from the infinite number contained in y = v (a, y) ; by giving 

 one curve, we select one surface from the infinite number contained in f(x, y, %, p, q) = 0. 

 Through any two curves can be drawn one or a definite number of developable surfaces : the 

 developable drawn through two curves of a surface answers to the chord drawn through two 

 points of a curve. Two points infinitely near determine a tangent : two curves infinitely near 

 determine a tangent developable, which (and not the tangent plane) is the correlative to the tan- 

 gent of a curve. When in two dimensions we have an ascending succession of notions A, B, C, 

 and in three dimensions a corresponding succession P, Q, R, S, the new notion which change 

 of dimension introduces is not S, but P : and A, B, C, answer to Q, R, S. Thus a polygon 

 presents angular points, sides, and area; a polyhedron presents corners, edges, faces, and 

 volume : of which angular points answer to edges, sides to faces, area to volume ; the corner 

 being a new introduction for which space of two dimensions has no analogue. The angular 

 point of a plane angle has its correlative, not in the angular point of a solid angle, but in the 

 containing straight lines : and generally, of correlatives in two and three dimensions, the 

 second is a dimension higher than the first. 



To a curve described by points answers what I have called a shaded surface, a surface 

 covered by curves according to a law. The curve which has the tangent y = ax + fa is 

 determined by making a self-compensating. If we shade the surface » = (a?, y) in the 

 manner dictated by y = \j/ («, a), we obtain the developable which touches the surface in the 

 line of shading belonging to any value of a, by eliminating x, after substitution for y and x, 

 between £ - * = p (f - *) + q ( 9 - f/) and (r + s\|/ T ) (£-«)+(«+ tyj (17 - y) = 0. If in 

 the first of these we make a a self-compensating variable, assuming x a function of a derived 

 from the second, we have p a (£ - x) + q a (q - y) = 0. If * = qj be itself not a developable 

 surface, this third equation cannot consist with the second, except upon the condition £ = x, 

 r) = y: for any other supposition gives p I{y q a - y I[1/ P a = 0, or q = fp, making y = developable. 

 Hence £ = (p (f , rj), the original surface, is the connecting surface of all the developables which 

 touch it in curves. 



Constructive reasoning shows that by assigning two points we assign one, or a definite 

 number, of the curves in y" = ^ (x, y, y) : when the two points are infinitely near they deter- 

 mine y , and y" = ^ determines the curvature. Let us now suppose we know, as to a surface, 

 one curve, the tangent developable through that curve, and the equation 



f(x,y, !«,p, q, r, s, t) = 0. 



Let the curve be one line of shading, and taking its projection on wy, draw successive projec- 

 tions for other lines of shading, each infinitely near the preceding. On the given curve, the 

 tangent planes are known from the developable, whence p and q are known : the curvature of 

 the curve gives one relation between r, s, t, throughout the curve ; the equation q = Fp, or 

 dq = Fp . dp of the developable gives another ; and / = is a third ; whence r, s, t, are known 

 on the curve. [Or thus ; — if the curve give y = ax, x = fix, then fi'x = p + qa'x, which, com- 

 bined with q = Fp gives p = fix, q = vx. Hence y!x = r + sa'x, v'x = s + tax, which, with 

 / = 0, determine r, s, t, in terms of xj. Hence we can establish, with infinitely small errors 



69—2 



