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

 F.E.SS.L.SfE., F.C.P.S., late Fellow of Queens' College, Canfih^idge; Pro- 

 fessor of Mathematics, SfC. in the University of Edinburgh. 



(Read 20th January 1840.) 



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

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

 general, point of view. I endeavoured 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 their 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 kind. 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 o-e- 

 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 with 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. 



