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XXIX. — On General Differentiation. Part II. By The Rev. P. Kelland, M.A., 

 F.R.SS.L.SfE., F.C.P.S., late Fellow of Queens" College, Camh-idge; Pro- 

 fessor of Mathematics, 8fc. in the University of Edinburgh. 



(Read 20th January 1840.) 



In a former memoir on this subject, it was my endeavour to exhibit the prin- 

 ciples of the science of General Differentiation in a simple, at tlie same time in a 

 general, point of view. I endeavom-ed to deduce, from one general formula, re- 

 sults easy of application in all instances ; and thus to exhibit the unity of the 

 different parts of the science, and the completeness of its fundamental formulae, 

 shewing at the same time the facility of theu- adaptation to particular and varied 

 cases. With the exception of certain expansions by means of a theorem analo- 

 gous to the series of Taylor, I gave no application of the principles to problems 

 of any land. It is my intention in the present memou- to supply this branch of 

 the subject, without which, indeed, however interesting may be the details, as a 

 portion of pure analysis, they will offer little to interest any but those who attach 

 themselves to the study of analytical combination. We hope, by the exhibition 

 of a few simple mechanical problems, solved by this process, to give to our sub- 

 ject an interest in the eyes of all, derived not from its intrinsic beauty, but from 

 its use as a medium of demonstration. It is well known that considerable diffi- 

 culty hangs over several very simple inverse mechanical problems ; from the ge- 

 nerality of their statement, a direct solution is sometimes impossible by the 

 ordinary methods. We shall shew that by our process such solutions are attain- 

 able Avith the greatest readiness. By this means we hope to give a value to our 

 subject as a branch of knowledge, independent of that value which it must pos- 

 sess from its curious and elegant structure. 



I must not conclude my introductory observations, without distinctly dis- 

 claiming the merit of having originally conceived the possibility of applying this 

 science to mechanics. M. Liouville has not only broached the method, but has 

 applied it to a number of cases in his first memoir. The theorem by which my 

 processes are effected is, however, as far as I know, quite new ; and one more 

 elegant or simple, considering its comprehensive nature, I can scarcely conceive. 

 But I proceed to its demonstration. 



