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XXVIII. — On General Differentiation. Parti. By The Rev. P. Kelland, M.A., 

 F.R.SS.L.SfE., F.C.P.S., late Fellow of Queens' College, Cambridge; Pro- 

 fessoi' of Mathematics, Sfc. in the University of Edinburgh. 



(Read 2(1 December 1839.) 



We owe to Leibnitz the iii-st suggestion of Differentiation, with fractional 

 and negative indices, but no definite notion of the theory was attained until 

 EuLER expounded it in the Petersburgh Commentaries for 1781. Still Eulek 

 wrote only a few pages on the subject, so that the theory could scarcely be said 

 to have come into existence, until Laplace, in his Theorie des Probabilites, and 

 Fourier, in his Theorie de la Propagation de la Chaleur, shewed how general dif- 

 ferential coefficients might be deduced by means of definite integrals, provided we 

 assume or prove, by means of some elementary definition, that the differential 

 coefficient of a circular or of an exponential function has a certain form. The 

 formula given by M. Fourier is a very simple one ; and our astonishment is 

 great, when we reflect on the time which elapsed from its announcement to the 

 first application that was made of it. This took place in 1832, in a memou- by 

 M. LiouviLLE, entitled Questions of Geometry and Mechanics resolved by a neir 

 analysis, which memoir is followed by two others on the more immediate theory 

 of the analysis itself. Although M. Liouville regards his analysis as a new 

 invention, we have no doubt that the idea is due to Fourier ; but still to M. 

 Liouville belongs the honour of moulding it into a shape capable of being made 

 use of in the solution of problems. 



M. Liouville, in this memoir, adopts a different line of proceeding, in order 

 to deduce the differential coefficients of positive powers of x from that by which 

 he obtains the differential coefficients of the negative powers. It is true he shews, 

 in one or two cases, the possibility of deducing the one directly from the other, 

 by means of complementary functions or constant functions of differentiation. 

 We think, however, with Mr Peacock, who revieAved this memoir in the Reports 

 of the British Association, that the process is far from satisfactory. Indeed, M. 

 Liouville appears to have entertained some suspicion that it would be thought 

 so; for we find, in the eleventh volume o/'Crelle's Journal for 1834, a short me- 

 moir by him on the theory of complementary functions, in which he corrects his 

 previous memoir, by adopting a more enlarged definition of / n (Legend re's or 

 Euler's Function). But this correction does not appear by any means perfectly 



vol. xiv. part ii. 5 H 



