532 The Rev. R. Canniehael on the Singular Solutions 



where j9i= -—-, provided that they satisfy the given equation and 

 do not at the same time make 



ax 



Now upon this it may be remarked, that in the last and best 

 collection of examples on the higher branches of the integral 

 calculus which has been published, it is somewhat singular that 

 the instances which are cited as illustrative and confirmatory of 

 this theory, are all merely cases of Clairaut's theorem, in which 

 the singular solution is readily determined from the general 

 solution, and where, as the applicability of the theory appears to 

 be accidental, so the truth of the results obtained by its applica- 

 tion furnishes no verification : in fact, that as this method coin- 

 cides in all its steps with that which determines the singular 

 solutions by eUmination of the arbitrary constants from the pri- 

 mary solution, and which must be true, so the verity of the 

 results obtained by the theory above stated cannot be looked 

 upon as confirmative of either its adequacy or its generality. 

 But, more than this, in the cases cited in the justly-valued col- 

 lection of examples above referred to (Gregory's ' Examples on 

 the Differential and Integral Calculus,' edited by Walton, Cam- 

 bridge, 1846), which do not fall under the Clairaut type, the 

 theory appears to fail, and the characteristics of such cases of 

 apparent failure deserve attention. 



In proof of the statement now made, the examples proposed 

 in illustration of the theory shall be cited seriatim. 



(1) ocp^—yp + m =0. 



(2) y-]r{y—x)p + {a-x)p'^=0. 



(3) y^—^xxjp -I- (l+a;>2=l. 



(4) x^ + 2xyp + (fl^ _ a?>2 = q. 



(6) (1+J92)(y_^)2_«2^2^0. 



Now the equations numbered (1), (2), (8), (6), are severally 

 reducible to the forms 



(•^?-2/)+- =0, 



{y~a:p)^+{p^-l) = 0, 



