214 
HOMOGENEOUS EQUATIONS. 
and from this equation, from what has been above said, e 9 will 
disappear on division by some power of that quantity, e. g. e ne . 
But the effect of simply removing a factor is the same as that of 
simply replacing such factor by unity. Now to replace e nd by 
unity is the same as to replace e e by unity, and if we do this 
• 
simply, i. e. without changing ^ and ~ , (17) will become 
„A dz , d?z dz\ n , 
F [ 1 ’ z ’ 29 + *’ dff 2 + T9J ~ 0 ( 18 )‘ 
. . x dz . d?z du du . 
Assuming then ^ = u, whence ^ ^ , we have 
tt du , 
^(1, z, u + z,u ~ + u) = 0 (19), 
an equation of the first order, which by integration gives 
u = (f>(z, c) (20). 
dz 
Then since u = ~, we have 
<f> (a, c) 
W-w^ + c' (21), 
J <j> (a, c) v '* 
in which, after effecting the integration, it is only necessary 
to write 
9 = log x, z = - (22), 
The solution of the proposed equation is therefore involved 
in (20), (21), (22). 
Ex. Given nx H — (y — x • 
Substituting as above x = e 9 , y = e e z, we find, as the trans 
formed equation, 
ftfz , dz\ (dz' 
U {29* + d9J~ \d9y