1887.] A. Mukhopadhyay — Differential Equation to all Conies. 139 

 which, being substituted in the equation hhv^(\ — ZX)^ = \ transform 



,2 



it into P(2i*= - ZvY = v - -, 



o 



which may be written 



27^V- 3(1-1- 12Fw»)i^+w»(l + 12F^') = 0. 

 Solving this as a quadratic in v, we have 



18F?; = (l-fl2A;V)±(l + 12/{;V)i 

 Introducing a new constant m, such that 12;?;*m'^ =1) this may be 

 written 



which gives 



V = ^ = I i {u^^-m^)±m(y-\-m''f \ 



%dx = 



Let u = m tan </>. 



Therefore 



m(l± cos 



2m di> 



Integrating, 



But 



2 cos^ - 2 sm"* - 



2mji? </) <l> 



■—X — l-ti = tan -, or, — cot -. 



o Z ^ 



2 tan 5 2 cot ^ 

 — = tan (^ = or, 



1 — tan'' - cot^ o ~ ^ 



; 2ma; 



I zmx \ 



(2inx \' 



Hence, 



log « =1 uax = m I --. --dx 



