BremiJcer — Errors Affecting Logarithmic Computations, 445 



S 0. 



For tliis purpose let us return to the series [3] of § 4 and put 



g— e/— 1 fop ^n ^ go that if i is written instead of y'— i they 

 are changed into 



-n.t,zl _^ ^ -(n-1) ct.zi^^^ ^^^e^. . . ^e(^-l)^x^^^^^«i^t = ^^(^) 



e y -\-e ^ ' v +.. .+ e".. .+e^ v +e v = cp (z) 



By multiplying together these series we have, as before, a series 

 whose coefficients show how many times the exponents can be 

 combined into any given sum. We have accordingly 



(Pi{z) . q>2{2) q)^ (^) = ^kt eizi 



if ^t is the coefficient of the power ^^^* in which t receives all 

 values from 



— n(ai -t- ^2+ +a^ ) to -{-nicr^ + «2+ a^, ) 



If we multiply this equation by e~^'^^ and integrate between the 



limits — 7t and + tt we shall have on the right hand side the sum 

 of integrals of which each one, as 



k^ i e dz 



becomes equal to zero if t and T have dl:fferent values, excepting 

 only the one in v\-hich t = t% which will equal 27i kt so that we 

 have 



jr— 1 cp^[z).(pJz) (p^{z)G ^^^dz — ki 



2it 



— TT 



[S] 



