REPORT ON CKRTAIN BRANCHES OF ANALYSIS. S19 



and have been chiefly directed to meet the difficulties connected 

 with the estimation of the values of the coefficient of differen- 

 tiation in the case of fractional and general indices. If we 

 should extend those investigations to certain classes of tran- 



which is easily reducible to the form, 



df^l ^ (-irr(i + ^) 



an expression which we have analysed in the note on p. 211. This part of 

 M. Liouville's theory is evidently more or less included in M. Fourier's views, 

 which we have noticed above. The difficulties which attend the complete 



developement of the formula — for all values of ft, which the 



principle of equivalent forms alone can reconcile, will best show how little 



progress has been made when the jttth differential coeflScient of — is reduced 

 to such a form. ^ 



M, Liouville adopts an opinion, which has been unfortunately sanctioned by 

 the authority of the great names of Poisson and Cauchy, that diverging series 

 should be banished altogether from analysis, as generally leading to false 

 results ; and he is consequently compelled to modify his formulae with refer- 

 ence to those values of the symbols involved, upon which the -divergenc)' or 

 convergency of the series resulting from his operations depend. In one sense, 

 as we shall hereafter endeavour to show, such a practice may be justified ; but 

 if we adopt the principle of the permanence of equivalent forms, we may 

 safely conclude that the limitations of the formulae will be sufficiently ex- 

 pressed by means of those critical values which will at once suggest and re- 

 quire examination. The extreme multiplication of cases, which so remark- 

 ably characterizes M. Liouville's researches, and many of the errors which he 

 has committed, may be principally attributed to his neglect of this important 

 principle. 



It is easily shown, if /3 be an indefinitely small quantity, that 



2/3 (m -(- m) /3 



and that consequently any integral function A -|- A a; -|- . . A„ xV, involving 

 integral and positive powers of x only, may be expressed by 2 A^ t"*^, where 

 TO is indefinitely small ; and conversely, also, 2 A^j e^^ may, under the same 

 circumstances, be always expressed by a similar integral function oix. M. Liou- 

 ville, by assuming a particular form, 



^^^ 2^ - 



where C is arbitrary, and fi indefinitely small, to represent zero, and differen- 

 tiating, according to his definition, gets 



^ = C v^ l^^^^EllZi' = c (l::i:^^£ll • 



_ _dx^ '^2^/3 2 ' 



but it is evident that by altering the form of this expression for zero we might 



show that was equal cither to zero or to infinity ; and that in the latter 



