218 THIRD REPORT 1833. 



partly for the purpose of illustrating the principle of the per- 

 manence of equivalent forms in one of the most remarkable 

 examples of its application. The investigations which we have 

 given have been confined to the case of algebraical functions, 



cients of algebraical functions, through the medium of their conversion into 

 transcendental functions by means of the very remarkable formula, 

 2 /*+ 00 /»+ CO 



<?•«= — ^ <p(«)d« / (p{a.)dqcosq{x — oi.), 



- 03 - CO 



which immediately gives us, 



6r (px _ 2 /*+ CO />+ CD (?»• 



~d^-lrj <^{«-)dciJ <P{cc)dq-^^r COS q(x-o,); 



- CD - 03 



which can be determined, therefore, if ^ cos q (,x — a.) can be determined, 



and the requisite definite integrations effected. If, indeed, we grant the prac- 

 ticability of such a conversion of (p («) in all cases, and if we suppose the 

 difficulties attending the consideration of the resulting series, which arise 

 from the peculiar signs, whether of discontinuity or otherwise, which they 

 may implicitly involve, to be removed, then we shall experience no embarrass- 

 ment or difficulty whatever in the transition from integral to general indices 

 of differentiation. 



In the thirteenth volume of the Journal de I'Ecole Polytechnique for 1832, 

 there are three memoirs by M. Joseph Liouville, all relating to general in- 

 dices of differentiation, and one of them expressly devoted to the discussion 

 of their algebraical theory. The author defines the differential coefficient of 

 the order f<, of the exponential funetion e"""^ to be mf^ e'"'^, and consequently 

 the ^th differential coefficient of a series of such functions denoted by 2 A^ e*" * 

 must be represented by 2 A^ rtT e""'. If it be granted that we can properly 

 define a general differential coefficient, antecedently to the exposition of any 

 general principles upon which its existence depends, then such a definition 

 ought to coincide with the necessary conclusions deduced by those principles 

 in their ordinary applications : but the question will at once present itself, 

 whether such a definition is dependent or not upon the definition of the simple 

 differential coefficient in this and in all other cases. In the first case it will be 

 a proposition, and not a definition, merely requiring the aid of the principle 

 of the permanence of equivalent forms for the purpose of giving at least an 



hypothetical existence to ^ ^^ for general, as well as for integral values 



of ft.. M. Liouville then supposes that all rational functions of x are ex- 

 pressible by means of series of exponentials, and that they are consequently 

 reducible to the form 2 A^ e*"^, and are thus brought under the operation of 

 his definition. Thus, if x be positive, we have, 



— = / e-"*^' 

 X .J 



a. 



and therefore. 



d^\ r^ 





