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lY. 
NOTE OlS" THE APPLICATION OE SYMBOLICAL METHODS 
TO THE SOLUTION OE CERTAIN FUNCTIONAL EQUA- 
TIONS. By M. W. CEOETON. 
[Read April 25, 1888]. 
1 . Take tlie functional equation 
where A, a are given constants. Let <^ (^) = y ; 
e^^y = ay. 
If we operate on both sides by e^^, 
In like manner, e"^^^ - a^y ; 
and therefore f{e^^)y = f{a)y . . . (a), 
if / be any rational and integral function. Assuming it to hold for all 
functions ; e.g. for f {x) = log^, 
JiDy = \o^a.y] 
It is not pretended that this process gives the unique or complete 
solution : as if f{a) is irrational or logarithmic, &c., and therefore 
admits of several values, (a) is really equivalent to several equations. 
In fact, in the present example, suppose h = 3, and putting the 
question as one in finite differences, 
of which the complete solution is 
X 
