Equations of the First Order. 125 



many of which are soluble by ordinary methods. If wc apply 

 the first example to 



x+py=ap 2 , 



ft is j^j, f-'t = (^f, *-(<•+.*)*- £log(< + I + •fe). 



and the solution is 



py = tyt, t being - — £ — -, 



(1 +j» 2 ) * 



in which we have to substitute for p its two values 



y+ s/ip + ^ax 

 2a 



To each of these equations, or any case of them, the process 

 may be again applied, and so on to an indefinite extent. 

 Similarly, if we take y = xfp, solved by 



<*-+©• 



where 



we obtain, almost mechanically, solutions of 



y[a+p*fi= («x+py)f(p{a +/> 2 )~ i ) = (ax+py)\p, 



$p + a(xp - 2/) =pf{4>'p + ax) , 



with many other forms ; and all other known soluble forms may 

 be extended in a similar manner. 



A convenient mode of applying the process is to take any pair 

 of correlated functions u and v ; and to endeavour to form com- 

 binations of u, v y and w so as to obtain any given form. Thus, 

 taking 



