224 THIRD REPORT — 1833. 



lent forms, whethe.^ the process followed be direct or inverse. 

 Man)^ examples of the first kind may be found in the researches 

 of Poinsot respecting certain trigonometrical series, which 

 will be noticed hereafter, and which had been hastily gene- 

 ralized by Euler and Lagrange ; and a remarkable example of 

 the latter has ah'eady been pointed out, in the disappearance 

 of the functions with arbitrary constants in the transition from 



u to - — , when r becomes a whole positive number. The gene- 

 dx'' 



ral discussion of such cases, however, would lead me to an 

 examination of the theory of the introduction of determinate 

 and arbitrary functions in the most difficult processes of the 

 integral calculus and of the calculus of functions, which would 

 carry me flir beyond the proper limits and object of tliis Re- 

 port. I have merely thought it necessary to notice them in 

 this place for the purpose of showing the extreme caution 

 which must be used in the generalization of equivalent results 

 by means of the application of the principle of the permanence 

 of equivalent forms*. 



The preceding view of the principles of algebra would not 

 only make the use and form of derivative signs, of whatever 

 kind they may be, to be the necessary results of the same ge- 

 neral principle, but would also show that the interpretation of 

 their meaning would not precede but follow the examination of 

 the circumstances attending their introduction. I consider it 

 to be extremely important to attend to this order of succession 

 between results and their interpretation, when those results 

 belono- to symbolical and not to arithmetical algebra, in as much 

 as the neglect of it has been the occasion of much of the con- 

 fusion and inconsistency which prevail in the various theories 

 which have been given of algebraical signs. I speak of deri- 



* Euler, in the Petersburgh Ads for 1774, has denied the universality of 

 this principle, and has adduced as an example of its failure the very remark- 

 able series 



1— a»» (1— a>») {i — gm-i) (1 — a^i) (l - a*»-i ) (l — o>"-2) 

 1 — a + 1 — a2 + 1 - a3 + c, 



which is equal to m, when m is a whole number, but which is apparently not 

 equal to m, for other values of m, unless at the same time a = 1 : the occur- 

 rence however, of zero as a factor of the (m + I)**" and following terms in 





 the first case, and the reduction of every term to the form -^ in the second, 



would form the proper indications of a change in the constitution of the equi- 

 valent function corresponding to these values of m and a, of which many ex- 

 amples will be given in the text. 



