THE 



PHILOSOPHICAL MAGAZINE 



AND 



ANNALS OF PHILOSOPHY. 



[NEW SERIES.] 



NOVEMBER 1830. 



XLVI1I. On the Problems of the Calculus of Variations. 

 By Hugh Ker Can k hi en, Esq. M.A. Trin. Coll. Cam- 

 bridge.* 



HPHERE are few students who do not find difficulty in the 

 •*- study of the calculus of variations. This does not arise 

 so much from their want of familiarity with the notation made 

 use of, since this is nearly the same as that of the differential 

 calculus, but rather from their not having a distinct view pre- 

 sented to them at the outset, of the necessity for, and the ob- 

 ject of, the several operations performed. It will hardly be 

 denied that this is a fault of the method of Lagrange, when 

 applied only to the solution of the more simple problems. 

 His investigation contains the solution of the difficult as well 

 as the easier problems, and consequently a great part of the 

 process is unnecessary for the solution of the latter. For these 

 reasons it is here attempted, with the assistance to be derived 

 from M. Poisson's solution of the problem of the brachysto- 

 chron, contained in the first volume of his treatise on Me- 

 chanics, to give first a solution of the easier problems, -and 

 then to show how far this solution is applicable to the more 

 difficult, and in what way the solution of these may be com- 

 pleted. 



The object proposed is to investigate the relation which 

 the variables involved in a proposed function must have to 

 one another, in order that a definite value of this given func- 

 tion shall be a maximum or minimum. The most common 

 form in which this function is proposed, is the integral taken 

 between limits of an expression containing the variables 

 themselves and the differential coefficients of one of them 

 considered as a function of the other. The limits are some- 



* Communicated by the Author. 

 N.S. Vol. 8. No. 47. Nov. 1830. 2 T times 



