234. 



mined. And even after this first feeling of paradox is re- 

 moved, or relieved, by the consideration that the number of 

 the operations of known form is infinite, and that the opera- 

 tion of arbitrary form reappears in another part of the ex- 

 pression, as performed on an auxiliary variable ; it still 

 requires attentive consideration to see clearly how it is pos- 

 sible that none of the values of this new variable should have 

 any influence on the final result, except those which are 

 extremely nearly equal to the variable originally proposed. 

 This latter difficulty has not, perhaps, been removed to the 

 complete satisfaction of those who desire to examine the 

 question with all the diligence its importance deserves, by 

 any of the published works upon the subject, A conviction, 

 doubtless, may be attained, that the results are true, but 

 something is, perhaps, felt to be still wanting for the full 

 rigour of mathematical demonstration. Such has, at least, 

 been the impression left on the mind of the present wx'iter, 

 after an attentive study of the reasonings usually employed, 

 respecting the transformations of arbitrary functions. 



Poisson, for example, in treating this subject, sets out, 

 most commonly, with a series of cosines of multiple arcs ; and 

 because the sum is generally indeterminate, when continued 

 to infinity, he alters the series by multiplying each term by 

 the corresponding power of an auxiliary quantity which he 

 assumes to be less than unity, in order that its powers may 

 diminish, and at last vanish ; but, in order that the new series 

 may tend indefinitely to coincide with the old one, he con- 

 ceives, after effecting its summation, that the auxiliary quan- 

 tity tends to become unity. The limit thus obtained is 

 generally zero, but becomes on the contrary infinite when the 

 arc and its multiples vanish ; from which it is inferred by 

 Poisson, that if this arc be the difference of two variables, an 

 original and an auxihary, and if the series be multiplied by 

 any arbitrary function of the latter variable, and integrated 

 with respect thereto, the effect of all the values of that 



