370 Dr. C. J. Hargreave on Differential 



and differentiate; divide by 2cf, and expel c by observing that 



cf= —y=, and therefore cf = -y=- yi, and compare the differen- 



V92. y <£ 2 



tial equation thus arrived at with our given form, we shall ulti- 

 mately obtain the system 



d/jt, 1 /div . d^\ 



^~RV& +( ^ l ~d^) i 



dp, 1 / ' d&. , d^\ 



where -gt is not now unknown, but is equal to log [v + vV — 1). 

 The partial equation therefore becomes an equation of condition, 

 denoting that the solution is of the form suggested ; and the 

 system last above written determines /jl or/. Example 5 above 



is an instance of this ; and //, will be found to be — ~ log log y, 



and /is (logy) - *. The reader may ascertain that the method is 

 applicable to 



Ex. 12. p* + (a- +b¥-+ —)p + e=0 (Boole, p. 135), and 

 \ y x xy / 



find p and /. 



Equations of this kind are not uncommon, a circumstance 



which renders it often worth while to try if --p= be a possible 



argument for the solution of the partial equation. 



15. Another particular case deserving of special notice is that 

 in which <£ 2 = 1, each value of p being the reciprocal of the other. 

 If, moreover, <f> Y be symmetrical with regard to x and y } the par- 

 tial equation will be so also, a circumstance likely to suggest a 

 solution of it. When ^ = 1, P and Q are most easily found 

 from 



p - Q = (4-4>^ +i >- 



Take, for example, the equation 



y* + x* + a*-4>bxy 



we find 



P-Q=(2£ + ])- 



xy — b(x <2 + y^-\-d 2 ) 



