452 Cambridge Philosophical Society. 



also obtained by making a a function of x. It also easily gives a 

 conclusion arrived at by the author in his last paper, namely, that 

 when ^ =00 is satisfied and not y'=x> it follows that x*+X«X ^^ 

 infinite. 



7. The author gives his own version of the demonstration of a 

 theorem of M. Cauchy, for distinguishing extraneous and intraneous 

 solutions. If y =P, P being a given function of x, satisfy y^=-x{x, y), 

 that is, if P' and ^(a;, P) be identical, then y=P is an extraneous or 

 intraneous solution of 2/'=xGt, ij), according as 

 fP+P dy 



j: 



{x being constant) is finite or infinite for small values of jo. 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 diflferential 

 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'-^'lVy or y={x + aY, 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 (f(.T, y, a, b)=0, da, b<p=0, da, 4fr|j/=0) though 

 sufliciently general, is never shown to be so. 



Let y = iP(x, a, b), combined with y' = J/^., give a = A(x, y, y'), 

 b=B(x,y,y'), from either of which follows y"=x(x,y,y'). The 

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

 given curve, y=iax, may be made to solve this for some form oif; 

 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=-'nsx, 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 rt = A(a:, -mx, za'x) and b=B{x, ■usx, ■us'x), and write A{x, y, y') 

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



The equations rfA = Ay (y"—x)rfa', dB = Byi(y"—-)()dx are identi- 

 cally true. And Ayi = oo , or any relation which satisfies it, is a 

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

 that is, when y' appears in it. When Ayi = oo is satisfied by a rela- 

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

 ordinary solutions of Ay = oo are solutions oiy" = x\ but not (neces- 

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

 which makes Ay = oo may make Ay finite. 



Comparing A and B with ;//, we have 



Ay = - , "h , By= "h , 



