1893.] On Operators in Physical Mathematics. 131 





(119) 



Here the factor of x r is identical with /(r), by (84), which corrobor- 

 ates. 



58. Returning to (111), if we try to make a series for log a;, in 

 powers of x we obtain 





-log x = 2 al-+~. . . . /'(r) 



= 2 af /'M-/0-1) 4- ~ ./'(r-2)-i/(r-8) 4- . .. . .(120 



This is done by making r be the representative power throughout, 

 by reducing the value of r by unity in the second term in the first 

 series, by two in the third term, and so on. Or 



(121) 





This is striking, but not usable. 



Also, if we try to get a series for x~ l we fail. The property (84) 

 comes in, and brings us to x~ l = x~ l in the end. This failure is not 

 obvious a priori in factorial mathematics. 



Deduction of a Special Logarithmic Formula. 



59. Now let the formula (111) be specialized by taking r = 0. We 

 then have 



>/'(-!) +ar 3 /'(-2) + ....]. (123) 

 ere, for the negative values of n we have 



3)=]2, /'(-4)=-l3, (124) 

 and so on, whilst for the positive we have 



/'(o) = c, /'(i) = 0-1, /(s) = i(o-i-i), 



/W = |{c-(l +m ^...^)}, (125) 



K 2 



