Equations of the First Order. 373 



function of x and y } we have 



+ |log(* 1 + R)rfy = 0, 



which becomes 



Dlog(jO + (&-B) 2 ))=0, 



if X be so taken as to make the other two terms vanish. Now, if 



c=t 1 be one solution of the first auxiliary equation, this condi- 



/dr \~ l dr 



tion is satisfied by making \= ( ~ ) , or log \= —log ~ ; for 



these two terms then become (if multiplied by fa+Jfy 



which, being 



vanishes identically if t, be a solution of 



'fy= (</>!+ R )^« 



Mongers second equation has therefore the two solutions, 

 ( R ^) (^ + r y(^i-R))= instant, 



( R ^ 2 ) \> + 0(*i+ R ))= constant J 



t 2 being the other integral of the first auxiliary. The two first 

 integrals of the partial equation therefore are 



y+(0 1+ RJ 9 =2B 



These give 



P+(h+*)9=Z* d fx*Tr 





