3G0 Dr. C. J. Hargreave on Differential 



ru= - (^- — 1)'* ; or we may make p=\ogf, and 



or=log(^r + V^—\) 9 

 and so on. 



In this form of the primitive p or -j- has the two values, 



(dp dm\ /dp dm\ 

 fa + d^r\fy + fy)' 



(dp dm\ /dp dm\ 

 dz~dx)^\lfy~dy/' 



which therefore by hypothesis are the roots of 



This gives the two equations 



p 1 +p 2 =2(j> li 



P\P*=<t><l'> 



or 



/ '(dp\* _ /fe\ 2 \ dp dp __dm dm _ 

 * l \\dy) \Sy) J^JyTx dy -doc ' 



/(dp\* fdmV\ (dp dp dm dm\ _ . 



which may be put into the symmetrical form 



da/ dp , d/A dm /dm ^\_r\ 



dyKdx' + ^ %/"" ^T'V* +01 %/" ' 



TA* i Tx+^T y ni^v^ +^dyr°> 



Let 



dm ^_dp _ dm m dp _ dp ^_dp __ 



dy ' dy~ yi dx ' dx~ *' dy ' dx~~ 



These equations then are 



(l +&*}- ^fr +0 A*) «<>, 



Eliminate £. and ^ successively, and we obtain (E 2 being written 

 for fo 2 -^, 



(«-i)((*i+^) g -^4)=o. 



Now neither /J — 1 nor £* — 1 can be zero, inasmuch as m— p 



