138 



intraneous solution of y' = x(x, y), according as 

 ,p+ P dy 



£ 



xO>y)-xO> p ) 



(x being constant) is finite or infinite for small values of /3. This 

 theorem has attracted little notice in this country: the author 

 believes it to be fully demonstrated, and considers it one of the most 

 remarkable accessions of this century to the theory of differential 

 equations. 



8. It is observed that the validity of the extraneous solution may 

 depend upon the interpretation of the sign of equality by which A=B 

 is held satisfied when both sides are 0, or both infinite, even though 

 A:B=lisnot satisfied. Thus y'=2Vy or y = (# + ff) 2 , has the 

 extraneous solution y=0, which, however, is not a solution if by 

 y' = 2 Vy we understand in all cases y' : Vy=2. 



9. The common mode of obtaining the singular solution of a bior- 

 dinal (by combining <p(x, y, a, b)=0, d a , b<j>=0, d a , b<px\y=0) though 

 sufficiently general, is never shown to be so. 



Let y=\p(x, a, b), combined with y'=\p x , give «=A(a?, y, y'), 

 b=B(x,y,y'), from either of which follows y"=x(. x >y >!/'}• T ne 

 most general primordinal is /(A, B)=0, /being arbitrary. Any 

 given curve, y=vsx, may be made to solve this for some form of/; 

 but, generally speaking, this solution will be extraneous. For A and 

 B are so related that every intraneous solution makes A and B„ con- 

 stant. And any primordinal equation whatever may in an infinite 

 number of ways be thrown into the form /(A, B)=0, so that the 

 intraneous solutions shall make A and B constant. 



(Given y=txx, required a key to all the primordinals of which it 

 is a singular solution. Take any equation y = \p(x, a, b), eliminate x 

 between a= A(x, "usx, ix'x) and b=B(x, ivx, ts'x), and write A(x, y, y') 

 and B(x, y, y') for a and b in the result.) 



The equations dA= Ay (y"—x)dx, dB=B y > (y"— x)dx are identi- 

 cally true. And A y > = oo , or any relation which satisfies it, is a 

 singular primordinal of y"=x> whenever it is a primordinal at all ; 

 that is, when y' appears in it. When A y > — oo is satisfied by a rela- 

 tion void of y', that relation is not necessarily a solution. The 

 ordinary solutions of A y > = oo are solutions of y"=x > but n °t (neces- 

 sarily) the singular solutions. The singular solutions of a relation 

 which makes Ay = oo may make Ay finite. 



Comparing A and B with \p, we have 



A/ = — ^ h B ■ — ^ a 



%y> = {log(W&r— ^o*)},- 



From these are obtained results in complete analogy with those 

 for primordinal equations. But when ^ipto— i/^o# = 0, the usual 

 criterion of singular solution, is made valid by \p n =0, \p b =0, a sin- 

 gular primitive of the singular primordinal may be obtained, which 

 does not necessarily satisfy y" = x- 



