448 Prof. Cayley on Differential Equations and Umbilici, 

 we have 



-^{(2Q'a+Qn') 2 -4r /2 n} 



= (2m-l) 2 A 1 A 2 (B ] B 2 r-y{ 2 /(/-.l)+2^ i ;} 

 = ^(2m-l)Y m - 1 [y <2 + (2m-l)x^] m - 2 \y(p^-l) + 2mxp} t 

 So that, the derived equation being 



Q 2 {(2Q'D+QD0 2 -4F 2 n}=O, 

 this is 



tyUy*™- 1 \y*+ fim-l^™- 2 \y{p' 2 -l) + 2mxp} =0. 



Hence, besides the factor Q 2 corresponding to the nodal curve, 

 and the factor □ corresponding to the cuspidal curve, we have 

 the factors y im ~ l and iy' 2 +(2m—l)xH m ~ 2 ; and, rejecting all 

 these, the differential equation in its reduced form is 



y{p*— l) + 2mxp = ; 



and the required verification is effected. The occurrence of 



as a factor in the complete derived equation would give rise to 

 some further investigations, but I will not now enter on them. 



I remark however that if m = l, viz. if the integral equation be 

 const. = x + \/x 2 + y 2 j or say z = x + \/x 2 -hy 2 j or, what is the 

 same thing, 



z 2 — 2zx—y q =0, 



then pbserving that t/ 2 + (2m — l)x^ is here =x*-{- which is 

 = □, so that □{2/ 2 +(2m-l)tf 2 }™- 2 =n.D- 1 = l, the dif- 

 ferential equation in its complete form is 



y(p*y + 2px—y)=0; 



so that we have here the factor y which divides out. The last- 

 mentioned result is most readily obtained directly from the 

 equation 



0=Q 2 (2Q'D +QD / ) 2 -4F 2 n=0, 



which is the derived equation corresponding to the integral 

 equation ^ = P + Q\Zn« We in fact have P=#, Q=l, 

 □ =# 2 + ?/ 2 , and the derived equation thus is 



(x + yp)*-(x*+y*)=0, 



that is, y(p' 2 y + 2px—y) = 0. 



I mention also, in connexion with the foregoing investigation, 

 the integral equations = x + \/2x 2 — y* 9 or z*— 2zx— x 2 -\-y 2 =0, 



