3G8 Dr. C. J. Hargreavc on Differential 



Therefore if the coefficient of tft divided by that of %" t> is a func- 

 tion of t (which call \t), then t is a possible argument ; and yt 

 is equivalent to §€~^ f dt, \t being ^\tdt. 



Suppose, for example, we desire to know what must be the 

 constitution of </> 1 and (f> q in order that is or /a may be a function 

 of xy. It will be seen that they must be such that 



2^ + Py + Qg 



is a function of xy. 



v q 

 Ex. 10. Thus, taking $! = #, <£ 2 = ~ } we find 



*=H«+T> *=-l»H) 



+ 



2y 



„2 > 



and the expression above written becomes (xy)~ l . Consequently 

 fM= log (xy), from which or and the primitive can be found. 

 Again, the condition that one of the two ; -cr and ^, may be a 



function of - (or t) is 



* x v ' 



,9 n . P a — - = a function of t. 



t — <C<p t + CpQ 



Thus if $i=a, <£ 2 = ^ this condition is fulfilled ; and we find 



1 



t • 

 Therefore 



^=7 + ^—72' 



cj=sin~ 1 

 and since 



r^V?4 



/&=log(#y), 



the complete primitive is found. 



In this example we cannot directly rationalize the primitive 

 into a quadratic in c ; but if we take the forms 



„ 1 2ax 



J cf y' 

 /being ^e- s ' m ~ 1 ^, and,/' being xye*' m **, we have by multi- 



