362 Dr. C. J. Hargreave on Differential 



The differential equation being 



p^-2cf> ] p-}-6 2 =0 } 

 and the primitive being 



c 2 -2^c + ^ 2 =0, 

 it is easy to find / and % so that the primitive may admit of the 

 form (v being (^j-f-^) 



in which case p and (c/+ %) c£j are identical. This is 



c^-2cf\ + \^-(v 2 -l)=0 (X being v- x ), 

 which (again altering the form of/) may be written 



c/nowbeins ,/^ + i(^r* +x ) ; 



or, finally, 



c*P-2cfylr+A = 0, where c/=f (l + \{-y=- -vj)> 



and (calling ^=fjj) 



V<k 



^-^(i+stfc-)> 



V# 2 



Now differentiate, divide by 2c/, and eliminate c by writing for 

 cf its equivalent, and we have 



or 



Comparing this with the given form of the differential equation 

 we have, tn being log (i^ + v^ 2 — 1)? 



dx dy ^ dy {^—\fdy dy 



< P 1 ^ + ^^-^ V ^ 2 ( ^"-(^_ 1) *^--- i1 ^ 



which are the same equations as before obtained, but without 

 irrelevant factors. 



