820 THIRD REPORT — 1833. 



scendental functions, such as e^", sin m x, and cos m x, we 

 shall encounter no such difficulties, in as much as the differen- 

 tials of those functions corresponding to indices which are ge- 

 neral in form, though denoting integral numbers, are in a form 



case the critical value infinity might be merely the indication of the existence 

 of negative or fractional powers of x in the expression for , which were 



not expressible by any rational function of e^^ under a finite form and in- 

 volving indefinitely small indices only. And such, in fact, would be the re- 

 sult of any attempt to differentiate this exponential expression for x or its 

 powers, with respect to fractional or negative indices. It has resulted from 

 this very rash generalization of M. Liouville that he has assigned as the ge- 

 neral form of complementary arbitrary functions, 



C + Ci a; + C2 a'2 + C3 a;3 + &c., 

 which is only true when the index of differentiation is a negative whole 

 number. 



Most of the rules which M. Liouville has given for the differentiation of 

 algebraical functions are erroneous, partly in consequence of his fundamental 

 error in the theory of complementary arbitrary functions, and partly in 

 consequence of his imperfect knowledge of the constitution of the formula 



— --^ — ~*~ ^' , : thus, after deducing the formula 

 r(l+M — »•) ° 



dr . ]--r- _ (- !)'• . a"- . r (n + >■) 



... («^ + °). - 1 . 2 . . . (« — 1) (o a; + 6)« + '■' 

 dx^ 



which is only true when « is a whole number, he says that no difficulty pre- 

 sents itself in its treatment, whilst n + r is >0, but that T {n -\- r) be- 

 comes infinite, when w + r ■< 0, in which case he says that it must be 

 transformed into an expression containing finite quantities only, by the aid 

 of complementary functions ; whilst, in reality, T (n + r) is only infinite 

 when 71 -\- r is zero or a negative whole number, and the forms of the com- 

 plementary functions, such as he has assigned to them, are not competent to 

 effect the conversion required. In consequence of this and other mistakes, 



dr \ 



in connexion with the important case {ax -\- b)'* , nearly all his conclu- 



dx'- 

 sions with respect to the general differentials of rational functions, by means 

 of their resolution into partial fractions, are nearly or altogether erroneous. 



The general differential coefficients of sines and cosines follow immediately 

 from those of exponentials, and present few difficulties upon any view of their 

 theory. In looking over, however, M. Liouville's researches upon this sub- 

 ject, I observe one remarkable example of the abuse of the first principles of 



, , r„, T r d^ cos ms ... , 



reasonmg m algebra. There arc two values ol j — , one positive and 



d x^ 



the other negative, considered apart from the sign of m, whether positive or 



1 ,3 



negative : but if we put cos m .r = — cos m x + — cos »i x, we get 



dr cos m x 1 d- cos mx . i. d^ cos m x ^ 



dx- dx- ax 



