501 



for the coefficient of t n in the development of any function of e l such 

 as/{y), in which are included, as particular cases, the two functions 

 treated in Dr. Brinkley's paper. In the case of (e* 1)*, n being 

 positive, the development so obtained agreed in form with that 

 arrived at by Brinkley, and before him by Professor Ivory ; but in 

 that of (e f l)~ n it differs from Brinkley's totally in point of form 

 (though affording, of course, the same numerical results), being 

 much simpler in expression and far more easily reduced to num- 

 bers, Neither was it at all apparent by what mode of transforma- 

 tion it was possible to pass from one form to the other ; and this has 

 ever since remained a difficulty. 



The essential difference between the two forms is, that in the gene- 

 ral coefficient, as expressed by Brinkley, the progression of terms of 

 which it consists are multiplied respectively by the successive differ- 

 ences of zero, 



A 3 0*+ 3 , &c., 



which run out in a diverging progression to infinity ; so that the 

 number of terms of which the coefficient consists is limited, not by 

 this progression coming to an end per sc, but by relations of another 

 kind ; whereas in the coefficient resulting from the other mode of 

 treatment, the differences of zero involved form the progression 



AO*, AV, ..... A*0*, 



which terminates per se at its #th term. A theorem subsequently 

 demonstrated by the author of this paper, however, in his " Collec- 

 tion of Examples in the Calculus of Finite Differences," affords the 

 means of expressing any term in the former progression by a series 

 of terms belonging to the latter. By substituting, then, the values 

 so obtained for each of those which occur in Brinkley's series, the 

 transformation in question is accomplished ; and the process, which 

 has the appearance of considerable complexity, is singularly simpli- 

 fied by the self-annihilation of all its most unmanageable terms. 



