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XX. — On General Differentiation. Part III. By the Rev. P. Kelland, AI.A., 

 F.E.SS.L.6fE., F.C.F.S., late Fellow of Queen s College, Cambridge ; Professor 

 of Mathematics, 6(c., in the University of Edinburgh. 



(Read December 21, 1846.) 



Nearly six years ago, I presented to the Society two Memoirs on the subject 

 of Differentiation, with fractional indices. The method which I adopted to extend 

 the signification of a differential coefficient consisted in assuming that the func- 

 tion / , which enters into the value of the coefficient deduced from a particular 

 hypothesis, is limited only by the definition » + 1 = » n. This generaUzation ap- 

 pears to be perfectly satisfactory, and promises to offer, if not the only, cer- 

 tainly the best extension of the Differential Calculus. Considering the length of 

 the interval which has elapsed since the publication of my former Memoirs, it is 

 remarkable that so little addition has been made to our knowledge of this branch 

 of analysis. With the exception of one or two papers in Liouville's Journal, 

 and a few remarks by Professor De Morgan, in his Treatise on the Differential 

 Calculus (pp. 598-600), I am not aware that anything has been written on this 

 subject since that time. Seeing, therefore, that others are not willing to enter on 

 this very promising field, I consider it not improper that I should make known a 

 number of extensions of this science to which I have been subsequently led, many 

 of which have been in my possession a considerable time. 



I must premise, that the object of this generalization of the differential cal- 

 culus is, not only to extend the bounds of research beyond the limits of that 

 science, but also to group and classify the results of the science itself It is, 

 perhaps, as important in the latter aspect as in the former ; for its very first conse- 

 quence is the union of the elementary forms of the two separate branches of that 

 science — the differential and the integral calculus — into one, so that the integral 

 becomes simply the negative differential. Now it is evident that this can only 

 be done by extending to some form, which is general for the existing calculus, a 

 universal and unrestrained interpretation. Such a form, properly selected, be- 

 comes, in the new science, a defining property, precisely in the same way 

 that the common differential coefficient is the defining property of the differen- 

 tial calculus. There are several forms which might appear appropriate to this 

 purpose : that which I have adopted is the differential coefiBcient of j:". The 

 assumption, therefore, on which the science is based, is the following: that 



VOL. XVI. PART III. 3 p 



