DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 23 



d , . d . 2a h ri 



(a (J) (a (!) + =0 



d,c v dy v y + ^ 



_ rf . /i ^\ tl ,1, f\ , 2/t-i-/ 2l-m-n 





Eliminating the term in 21 m n from the second and the third of thesu by 

 means of the original differential equation, we obtain the modified equations in the 

 forms 



A (h _ /} _ A. (h _ f} + 1 l=Ar^ = 



clx v dy^ J ' y + z 



d , . d , . . n k + f/ c 



(a c) -- : - (a c) + - = 0. 



dx vy dy w y + s 



By the first of the former and the second of the latter, we find 



Now, as j> = f/, 2 behaves like a constant under the operation c//<7^ djdtj ; hence 

 we have 



77 ; ) l ~ i = ' 



when 



^ - ^ , - 



7 / * " 



y / 



Consequently 



a -2;/ H- c , .. 



= arb. in. of x + ?/, 



when 



2 = arb. fn. of x -\- y; 



and we therefore infer that 



a '2(i + c 



y + 



. , > 



= <h(jc -{ 11, z) 



is an integral that can be associated with the original differentia] equation, <l> being 

 an arbitrary function. 



In order to proceed to the primitive, we take fii'st 



a '2y + c = (y + z)tl> (x + y, z), 

 and introduce a new arbitrary function f, defined by 



9 = Jm ~ 3/i u + 3/122 ~~ Jw-2> 



