Prof. Cayley on Differential Equations and Umbilici, 445 



is obtained from the general equation by writing therein 



5=0, c=l, #=0, f=zm; 

 we have therefore 



v{v 2 + 2m-l) = {v-u){v-l3){v-y), 



or say 



a=0, /3=i\/2m—l, 7=— i\/2m— 1; 

 and thence 



v*— 1 1 1 2m v 



v{v* + 2m— 1) 2m— I v ' 2m-l v 2 + 2m— 1 



^ v + i\/2m—l v—i\/2m—\ 3 

 giving 



2m— V 2m— 1 



The integral equation thus is 



2m 



const. =(m#+\/n)~ 1 (w# + v / n) 2m ~ 1 



m-l 



{(m#-f-2\/2»i — ly + \/D)(w^— W2m — 1 «/-1-v / D)S- 2wi ~^ 

 where □ =mV4y 2 ; or, observing that 



(m#-H\/2m— l2/ + \/C])(m#— i\/2m— \y + ^/~n) 



= (m«r + \/a) 2 + ?/ 2 



= 2m(mx <2 + y 2 -\- x\/ □ ), 

 the integral equation is 



l_ w-l 



const. =(ma7 + \/n) 2m ~ 1 (w^ 2 + y 2 + l rv / n) 2?re " 1 , 

 or, what is the same thing, 



const. = (mx + \/ Q )(m# 2 + y 2 + a?\/ Q ) m ~ S 

 the result given in the former part of the present paper. 



VI. 



I annex the following a posteriori verification of the solution 

 const. ==(mz + A>/n)(mx' 2 + y' 2 + x\/~n) m ~ 1 

 of the particular equation 



y(p 2 — l)+2mxp=f). 

 Putting for shortness 



A=mx+\/ Q, 



B = mx <2 + y' 2 + x\/n~ J 



