190 MR. ROBERT RAWSON ON SINGULAR 



where (/3) is a proper fraction (see Todhunter's Diff. Cal- 

 culus, p. 81, third ed.). Nothing as yet has been said 

 with respect to the conditional equation w — ^>{cc, y) by- 

 means of which (50) and (51) have been obtained. The 

 above general integral obtains whether w — o is a solution 

 of the differential equation or otherwise. Between the 

 limits zero and (iv) the integral becomes 



i 



• dw _ W Jo F ^^ 



From paragraph (13) it appears that F(<r, o) may be 

 and frequently is a function of x when w — o is a particular 

 solution. Boole's argument in supp. vol. p. 28, is not 

 universally correct. 



The inferences from (55) are as follows : — 

 When w = o and F(#, o) is a function of x, then 



i 



dw , ,. 



= ° (56) 



l/r [X, w) 



But whether w = o is a singular or a particular solution 

 the equation (56) does not determine. 



When w = o and F(x, o) is a constant, then 



f 



dw o , . 



• • • • (57) 



t/t (x, w) O 



This equation , which is indeterminate, gives w = o as a 

 particular solution if -y-F(<r, o) is finite. Boole's ex- 

 ample, supp. vol. p. 31, confirms the truth of (57) as log 



( - " '' ) , is indeterminate when y = o. There is nothing 

 VJogo/' y & 



whatever to show, in Boole's demonstration, that w = o is 

 a singular solution of the envelope species. 



