Equations of the First Order. 369 



plication a biquadratic in c, which includes also the solution of 



p *+2ap+ J =0, 



being in fact the primitive of 



If (j) l and <£ 2 are both functions of - or t, It is so also ; Qx 



and Yy, too, are functions of t ; so that the condition is satisfied. 

 This gives the solution of homogeneous equations of the second 

 degree. 



ar.ll, Let*,=|, fc=lg.-(»-l), U* = ^(J + n) 

 whence 



//I— 1\2 



This being to, fx is found to be I J log x ; and the primitive 



is 



— =2— . /hpine* a?' n ' 



c/+ -f =2^-, /being # v « ' (Boole, p. 130). 

 cj x 



We may go on thus indefinitely deducing special theorems for 

 any function of x and y. Thus //, will be a function of x + ay if 



- — - — ; s be a function of (x+ ay). 



The process, however, is tentative ; for the forms of <p l and cf> 2 

 do not generally supply any indication as to what particular 

 arguments can be employed with a probability of success. 

 14. Reverting to the primitive in the form 



(</+x) 2 -2<W+X) + <k=0, 

 we may observe that there is a class of equations which admit of 

 % being zero — that is, having a primitive of the form 



(c/) 2 -8&(c/)+&=0. 



Clairaut's equation, when of the second degree, is an instance of 

 this, and in it / is unity. 



If we place the equation in the form 



(cf)*-2v(cf) + l=0,;t> being -^k, 

 Phil. Mag. S. 4. Vol. 27. No. 183. May 1864. 2 B 



