IN THE THEORY OF DIFFERENTIAL EQUATIONS. 535 



consequent upon <t> = 0, ♦ = ; while, as to <p (tv, y, a, b) = 0, taken alone, the first equation 

 which is clear of constants is biordinal. That every surface dictates another surface, and a 

 mode of shading both, which makes each line of shading on one surface a polar reciprocal of 

 some line of shading on the other, is the nearest resemblance of properties which we can make. 



16. The simple theory of Lagrange for partial primordinal equations, is a case in which 

 the transition from constants to compensating variables is a considerable generalization. If 

 f{ai, y, z, p, q) — arise from elimination of a and 6 between (p (x, y, ss, a, b) —0,(p x + <p z p m 0, 

 <p v + (p 2 q = 0, a and b being constants, it equally arises when a and b are compensatory of 

 each other, as in <p a da + (p b db = 0. If b = Fa, and a be eliminated between <p = 0, <p a + <p b F'a = 0, 

 we have the general primitive, involving one arbitrary function. Constructive processes also 

 show that we cannot have more: the surface / = is determined so soon as one curve in it is 

 determined; just as the curve y = y\pe,y) is determined so soon as one point is determined. 

 This tempting analogy between the arbitrary function and the arbitrary constant has caused 

 much confusion, though it soon became evident that it breaks down as to biordinal equations. 

 Arbitrary constants, considered as compensating variables, are in all cases sufficiently general. 

 Hence Lagrange may be justified in calling (f> (x, y, z, a, b) = the complete solution of 

 f(x, y, ss,p,q) = 0: though it may seem strange that a = a (x + y) + 6 should be so called with 

 reference to p = q, when ss = F (x +y) does equally well for any form of F. But the two 

 forms are co-extensive when a and b are made compensating variables. Nevertheless, I prefer 

 to call (p (x, y, ss, a, b) = 0, a primary solution. 



The singular solution of (p(x, y, ss, a, b) = 0, or x = \J/ (<v, y, a, b), is obtained by eliminating 

 a and 6 from \^ = 0, 4r 9 = 0, <Ar t = 0, or <p = 0, - <p a '-(p z = 0, — (p b '.(p z = 0. That is, every 

 singular solution except those in which ss is absent, as in {x, y) = 0. Yi <f? ti% mQ, m =o, 

 give a = A(x, y, ss, p, q), b = B(), so that ss = \fs(x, y, ss, A,B) is identical with/(,r, y, «,p,g , ) = » 

 we see that the expressions written in the following lines are proportionals, 



^ + >M* + ^A' f, + *M + ^' - l + % A * + ^» B z> ^4. + ^» B P> ^a A 9 + ^B q . 



Jx Jy Jz Jp Jq 



Usually, then, f p = and f q = 0, or f z — oo , determine the singular solution. If M (x,y,ss,p,q) = 

 be the supposition on trial, we differentiate f = and M = 0, each with respect to x and to y 

 separately, and from the six equations we eliminate p, q, r,8,t; the relation (a?, y, ss) = pro- 

 duced, if a solution at all, is singular. 



The structure of methods of compensation (speaking of the first order only) is threefold. 

 First, when each variable is made self-compensating in every equation. When there is one 

 equation only, this method introduces an equation for every new variable, and gives singular 

 solutions. Secondly, when the new variables are made mutually compensative, and the number 

 of equations and variables is so related that elimination produces equations between the new 

 variables only, which can be integrated and new constants introduced. In this case the 

 forms obtained are of the same character as those which produced them. Thirdly, when 

 elimination does not produce equations between the new variables only, so that the only method 

 of satisfying the equations of compensation is by so many arbitrary assumptions of relation 

 Vol. IX. Pa#t IV. 69 



