1893.] On Operators in Physical Mathematics. 125 



of the highest repute could not see the validity of investigations based 

 upon the use of the algebraic imaginary. The results reached were, 

 according to them, to be regarded as suggestive merely, and required 

 proof by methods not involving the imaginary. But familiarity has 

 bred contempt, and at the present day the imaginary is a generally 

 used powerful engine, which I should think most mathematicians 

 consider can be trusted (if well treated) to give valid proofs, though 

 it certainly does need cautious treatment sometimes, and perhaps 

 auxiliary aid.* 



Application of Generalized Binomial Theorem to obtain a Generalized 

 Formula for log x. 



49. Let us now pass on to view the logarithm in its generalized 

 aspect. One way of generalizing logx is to regard it as the limit 

 of (djdn)x n when n = 0. Now, using the generalized binomial 

 theorem 



x r \n 



where r has any value and the step is unity, we obtain by this process 



(78) 



where the accent means differentiation to n, after which the special 

 values are given to the argument. Or, since 



<))' _ f(n) 



and ~ ~ 



\if(n) is the inverse factorial, therefore 

 But also 



= 1, /'(O) = C = 0-5772, 2 



by 17 and equation (94), Part I. So we reduce to 



log (l + s) = -C + 2 <f (r)/'(-r). (81) 



* Perhaps we may fairly regard the theory of generalized analysis as being now 

 in the same stage of development as the theory of the imaginary was before the 

 development of the modern theory of functions. Not that I know much about the 

 latter ; the big book lately turned out by Forsyth reveals to me quite unexpected 

 developments. 



