458 



By the aid of the symbol [^-j-] we may obtain another 



interesting development. In virtue of the equation (2) we 

 have 



hx — 



It is plain, then, that the symbol 



e '^' 

 operates on any function of x by changing x into e''ir ; that 

 is to say, 



(e'' x) = e '^'^ V (x) ; 



whence, developing the right hand member, we get 



ix—]F(x) («;7-) ^(^) 

 ^(ehx)=-F{x) + ^-^ h+ ^ [^ A2 + &C. (4) 



As Taylor's theorem gives the altered state of f (a;), after 

 X has received an increment A, so the theorem just announced 

 exhibits the new value of f {x) after x 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 



(4) it must be remembered that [x-7-j is not equivalent to 



x"^ -r-5 but tox-r- X -T-; and so on for the other powers of the 

 dx^ dx dx ^ 



symbol. Neglecting to make this distinction we should get the 

 development of f {x + xh) instead of f (e'^x). The actual re- 



d" d 



lation between the symbols x^ -r- and a; -- is obtained immedi- 



■' dx^ dx 



ately from the equation (2) which gives us 



