459 



Sir John Herschel has given the following theorem, which 

 enables us to develope f (e'') in a series of ascending powers 

 of A when such a development is possible : — 



F(e'') = r(l) + iF(l + A)o + j-g^Cl + A)o2+ &c. 



Comparing this with the one given above, we obtain the 

 following theorem : 



by the help of which we arrive at a still more general one, 



Sir William R. Hamilton wished to be allowed to remind 

 the Academy that he had communicated to them, in 1831, 

 another extension of Herschel's Theorem, which was pub- 

 lished in the seventeenth volume of the Transactions (page 

 236), namely, the following : 



v'/^(o')=/(i+A)v'(^(oO)''; 



where the accents in the first member might have been omitted, 

 and where y' denoted any combination of differencings and dif- 

 ferentiatings, performed with respect to o', and generally any 

 operation with respect to that accented zero, of which the sym- 

 bol might indifferently follow or precede /(I -{■ A), as a sym- 

 bolic factor. By making xp (o') = £»', and v' = d'^, where 



d'=: -— the theorem of Herschel is obtained. A much less 

 do'} 



general formula was cited as " Hamilton's theorem," in the 

 last Number of the Cambridge and Dublin Mathematical 

 Journal, namely, the following : 



f{x)=f(\ + A)x'; 

 which had, however, been also given in the same short paper 

 of 1831. 



