DEPARTMENT OF PURE MATHEMATICS. 489 



we can take 



.r' 2 — . 



The quantity -^^ is 1, and so does not vanish at all; and R is zero con- 

 currently with ^ = 0. In this example also, one hypothesis made in the course of 

 the general proof is not justified; all initial values that satisfy V' = constitute an 

 essential singularity for v, so that v is not a regular function in the vicinity of 

 those values. 



In both of these examples it should be noted that not merely is it impossible 

 to specialise the general integral so as to make ^ =/{»(, v), but also that the 

 equation ij> = is not part of any equation /(?<, v) = 0. 



As a further example, to illustrate the limitations of the hypothesis, consider 

 the equation 



obviously satisfied by 

 we can take 



^ = z-xy = 0; 



= i„^^-JL 



y =-»y 



^~ — 1—. = </.(=, «, r). 



uv — i-- 

 "it 



The quantity -^ does not vanish, either identically or in virtue of ^ = ; 

 and, in this instance, R is not zero. The explanation is that the argument, Icad- 

 ing to the relation R-^ =0, is based upon a supposition which is not justified ; 

 the equations 



u = u{x,y,z), v = v{x,y,z), 



cannot be resolved so as to express x and y as regular functions of z, it, v in the 

 vicinity of values that satisfy ■\\r = 0. 



Passing from the particular examples and returning to the interrupted argu- 

 ment, we have seen that the necessary relation will be satisfied if R = 0, concurrently 



with ^ = 0. But this condition gives no information concerning ^*^ ; and all that 



can be declared is that, if R = is satisfied concurrently with \/' = 0, the proposition 

 as to the specialisation of the general integral is not necessarily valid. The result 



has been obtained on the supposition that zC"^\ does not vanish identically. 



Had we proceeded from the supposition that j("'^) does liot vanish identically, 

 then the same doubt about the comprehensiveness of the general integral arises 

 for integrals ^^ = which make P = 0; and, similarly, with j(^), for integrals 

 1/^ = which make Q, = 0. 



Naturally we may assume that P, Q, R do not vanish together for an integral 

 •^•=0, because the equation would then be satisfied without reference to the deri- 

 yatives of s. We may neglect as trivia} those integrals ^ = (if any) of an 



