POINTS OF THE INTEGRAL CALCULUS. [Ill] 



The base-curve of the tangent cylinder is the singular solution of y = ^, to which all the 

 projections of the horizontal sections are ordinary solutions. When the tangent cylinder meets 

 the surface in an oblique curve, to every point of which a distinct horizontal section of course 

 belongs, then the singular curve is touched by a primitive curve at every point. If from any 

 point (x, y, c) of the surface, we proceed to another (x + dx, y + dy, c + dc) infinitely near to 

 it, the equation dc = <S>,dw + Q? y dy usually determines the ratios of dc to d.v and to dy in 

 terms of y . This connexion is momentarily severed when <b x and <b y , either or both, become 

 infinite. If we take advantage of this to assume dc in such manner that $, or <J> y or both (as 

 before) shall remain infinite when x, y, c, become x + d,v, y + dy, c + dc, and if we repeat this 

 process, we draw a curve through the vertical elements of the surface, being the curve of con- 

 course with the tangent cylinder. The projection of any point in this curve of concourse is a 

 point at which two projections have a common tangent, the two being those of the curve of 

 concourse, and of the horizontal section passing through the point. If the tangent cylinder 

 touch in no oblique curve, but only in horizontal sections, the singular solution consists of 

 primary solutions only, which have no contacts with others. But if the tangent cylinder not 

 only touch in an oblique curve, but also touch or cut in a horizontal section, then the singular 

 curve is itself a primary curve, touched by another primary in each point of it. And if the 

 curve of concourse have convolutions or branches, all oblique, each point of the singular curve 

 may be the point of contact with several different primaries. To consider only extraneous 

 solutions, to the exclusion of other singular solutions, is not to mark out, as objects of special 

 attention, those surfaces whose tangent cylinders have oblique curves of concourse, to the 

 exclusion of those which have them only horizontal. There is another, and a most unsystematic 

 exclusion : that of all cases in which the tangent cylinder touches in an oblique curve, and 

 also touches or cuts in a horizontal one. When <&* = oo, &c, gives only c = const., the curve 

 of concourse is only a horizontal section: but when it gives c = ^(x, y), and (p\x, y, \|/(a?, y)\ =0 

 is equivalent to <p {as, y, k) = 0, k being some definite value of c, there is an oblique curve of con- 

 course, and also a horizontal section. For instance, the primitive being y = x + c (x + \/c) 2 , 

 the branches of the singular solution are obtained from x + ^/c = 0, and x + 2-y/c = 0. The 

 first gives y = oo, the same as arises from c = ; the second gives the extraneous solution. But 

 both give curves of separation, of contact, and of resilience, to the primary curves. 



The following instance will serve to illustrate that when the singular solution is itself a 

 primary, which is neither touched nor cut by other primaries, and is nothing but a curve of 

 separation, it is still one of those extreme cases which analysis admits when it clearly sees the 

 way up to the limit. Let the primitive equation be 



oo> + mxy + ny 2 + (ax + by) c +p<? = q, 



m, n, &c. being fixed constants : and let the surface be an ellipsoid. Draw the vertical 

 cylinder which touches this surface, and let a plane parallel to ooy, beginning and ending as a 

 tangent plane to the ellipsoid, move so as to pass through all the intermediate horizontal 

 sections. It will then be obvious, that the connexion of the primitives and singular, in the 

 plane of xy, is as follows. The singular curve is an ellipse, in which are two points, say 

 A and B, similarly situated on opposite sides of its centre. The primitive is an elliptic wave 



