458 
By the aid of the symbol (5) we may obtain another 
interesting development. In virtue of the equation (2) we 
have 
het 
e Mtr (e" x)". 
It is plain, then, that the symbol 
d 
oa 
operates on any function of # by changing 2 into e*a; that 
is to say, 
hae 
F(e*x) =e." F(2); 
whence, developing the right hand member, we get 
F(eba) = F(a 2) +--+ &e. (4) 
As Taylor’s theorem gives the altered state of ¥ («), after 
x has received an increment h, so the theorem just announced 
exhibits the new value of F (a) after « has been multiplied by 
a number whose logarithm is h ; the series in both cases being 
arranged according to ascending powers of h. 
In executing the operations indicated in the development 
2 
(4) it must be remembered that («<) is not. equivalent to 
Pa dist 
ip a but to a — me wane and so on for the other powers of the 
symbol. Neglecting to make this distinction we should get the 
development of r («+ ah) instead of r (e*x). The actual re- 
lation between the symbols 2” = and x <. is obtained immedi- 
ately from the equation (2) which gives us 
eae = (tg) (ge 71) oe (ag 
eee haem oP erected ve —n+1). 
ee 
