126 Mr. 0. Heaviside. [June 15, 



50. To obtain the common formula for the logarithm, take r = 0. 

 Then, since 



we reduce (81) to 



= x + tf x + ..... (82) 



When r = I in (81), we have 



Now here all the differential coefficients of the inverse factorials 

 may be put in terms of /( i) by means of the formula 



/() + /'()= /(-!), (84) 



which follows from 



w /(0=/(w-i); r85) 



but since the resulting formula does not seem to be useful, and is 

 complicated, it need not be given here. 



Deduction of Formula for 

 51. If we differentiate (81) with respect to #, we obtain 



= 2 */(,)/' (_r-l), (86) 



where the second form of the series is got by increasing r by uniby in 

 the first. Here note that we have a definite expansion, whereas in 

 32 we found the binomial expansion to be indeterminate. When 

 r = in (86) we have, of course, the special form 1 x+ a? ..... 

 It is also right when r = |. 



Deduction of Formula for e~ x . 



52. Now regard (86) as true analytically, and we can obtain a 

 formula for e~ x . For, first put V" 1 for a?, giving 



