BURMANN'S THEOREM. 117 



and if a = 0, so that -fy (x) vanishes with x, 



3?d r 



*.r~ 



EXAMPLES. 



1. Given y = z + xtf, expand y in powers of a:. 

 Here <j> (y] = e", 



, 1 rt \4/ \ -7 / \T // / \ I VV * _1 *, 



therefore ^-^zi -j </> (*)| / ( s ) l = ^7 -i e = n e 

 Thus v = 



Ll Li I* 



2. Given y = z + x , expand y in powers of x. 



Here 



therefore -T-J 

 Hence ?/ = zH 



3. Given xy logy = 0, expand y in powers of x. From 

 the given equation 



y' = 



