416 Mr Hill, On the series for [May 10, 



The whole investigation being based on the Exponential 

 Theorem, the writer has developed a proof of this theorem dis- 

 tinctly indicated, but not fully set forth, by Sir W. R. Hamilton in 

 his Theory of Conjugate Functions or Algebraic Couples in the 

 Transactions of the Royal Irish Academy for 1837 (pp. 411 — 412). 

 As the writer has not seen this proof in any text-book, the Cam- 

 bridge Philosophical Society may consider it worth printing. 



The Binomial and Multinomial Theorems for positive integral 

 indices will be assumed to be known. 



Art. 1. The proof of the Exponential Theorem proceeds very 

 nearly after the manner of Euler's proof of the Binomial Theorem, 

 except that here the equation 



f(m)xf(n)=f(m + ri) 



where f(m) = 1 +m + ^y + ••• +^y + ••• 



is directly proved. 



Art. 2. The identity of the series 



(x x 2 x 3 \ x l (x x* , x s V, 



a 



n fxxx' 



+ h(i + 2+3 + -; + 



and the series 



n (n+1) - n (n + l) ... (n + r — 1) 

 1 + nx + — ~- — - x 2 + ... + — ^ -r of+ ... 



— ■ 1 ' • 



is established. 



Art. 3. Putting in the identity of the last article, n — 1 ; using 

 the Exponential Theorem, and the known expression for the sum 

 of a Geometrical Progression, it is shewn that 



1 — X 



whence T + 2 + "3 + '" =_ loge ^ ~ ^' 



which is the logarithmic expansion. 



Art. 4. Making use of the Exponential Theorem and the 

 result of the last article, the first series in Art. 2 may be written 



N _„ , n(n + l) ;, w(w+l) ...(n + r-1) r 

 .: (l-x) n =l + nx+-±Y l — V-I-...+ — 2i * + ": 



which drives the Binomial Theorem. 



