20 PROFESSOR A. B. FORSYTE ON THE INTEGRATION OF 



whence 



p = p\, ft = a + |, y = a + ~ , 

 so that 



A = a (.x + y + z) + ly + 2^ 2. 



It is easy to verify that, when p is distinct from and co , there is no intermediary 

 integral, that is, no relation between /, m, n, involving only one of the arbitrary 

 functions and <. We have 





and then the differential equation is 



It has no intermediary integral. Its primitive is 



v = <(> + e -i- \ {p (^ - ^) + ^ -- ^}, 



where 



^=^ + ^2), 5 = (9( a; + 2,?/); 



and X has the above value, and p is neither nor oo . 



11. Now take a particular case, so as to illustrate the method of integration. 

 Let 



a 0, p 1 ; 

 then 



* = i(yt*). 



The difierential equation is 



/, , , 11 m n 



a + f.-c,-h+ =0; 



U T ~ 



and it is required to obtain the primitive 



" = 4> + 0+i(2/ + *) W > 1 -<fe + 1 -0 2 ), 

 where 



^ = ^ (a; + y, z), 6 = 6 (x + z, y). 



For the differential equation thus postulated, the characteristic equation is 



p 2- pq+p - q = 0j 



that is, 



