TRANSACTIONS OP THE SECTIONS. 525 



only to individualized values, to formulae more or less indefi- 

 nite, containing those values among others. This is in fact the 

 paralogism of applying to an equivocal term used in one sense, 

 a predication proved only vpith respect to a different sense. He 

 adopts the position of M. Crelle, {Journal fur die reine und 

 angewandte Mathematik, torn. vii. cah. 3 and 4,) that no equa- 

 tion is admissible, of which one side may not be proved to be, 

 by previous consistent postulates, an " identical transformation" 

 of the other. He vi'ould not banish diverging series from ana- 

 lysis, but he agrees with M. Poisson and M. Cauchy in holding 

 that the remainder or complement of a series, even after an in- 

 finite number of terms, ought always to be taken into considera- 

 tion, since postponement, however long continued, cannot, of 

 itself, destroy. He goes so far as to maintain that, even in con- 

 verging series, this remainder, though an infinitely small quan- 

 tity, may, in certain cases, produce sensible effects. Thus, in 

 his opinion, we are not always at liberty to assume that the sum 

 of the series obtained by differentiating an infinite number of 

 terms of a converging development will approximate indefi- 

 nitely to the differential coefficient of the function, because (as 

 he shows by example) the differential coefficient of the infinitely 

 small remainder may be of finite magnitude. He assumes the re- 

 ceived symbolic rules of algebraic addition, subtraction, multipli- 

 cation, and division, (which are in accordance with certain lead- 

 ing and elect truths of numerical science,) and he proceeds in 

 like manner to define exponential quantities and logarithms by 

 means of properties which he supposes that mathematicians 

 would generally acknowledge to be characteristic and funda- 

 mental. He admits also the theorems of the integral and the 

 differential calculus as derived from the consideration of limits. 

 From these definitions and postulates, he contends that his 

 conclusions not only legitimately follow, but are consistent with 

 received notions, as far as the latter are consistent with them- 

 selves and with each other. 



He explains a-^ (where a and x may be any quantities, real 

 or imaginary,) by means of the following functional definition, 

 viz. "a* comprises in succession every function {<p x) of x, which, 

 independently of x and x^, fulfils the following conditions : 



(px 4>x' = <p{x + x')\ , J V 



<|> 1 = a" J ^ 



From this definition, (which Mr. Hamilton recommended him 

 to make, in explicit terms, the basis of his former Essay,) he 

 proceeds to evolve all the properties of a". It embodies the 

 well-known characteristic which led to the extension of expo- 



