8 REPORT — 1859. 



z as a function of ec and x, and x as a function of ec. For the total derived func- 

 tions, therefore, we shall have the values 



(dV\dV , dV d f. + d ^( d ± + dz dx\ 

 \dcc) dec dx dec dz\dec dx dx)' 



(d]h\ = d]h + %dx^ 

 \d») da, dx dec' 



\dccl dec 



dy 2 dx 

 dx dec 



By introducing these values into the preceding expression, and afterwards re-esta- 

 blishing the symbols of substitution before withdrawn, we arrive at the symbolical 

 formula 



rf.DV 



dec 



I \ dec dx 'dec dz \dec dx dec) f 



Vi 



*2 Vi Z-2 x i V\ *a 



+ 



\dec dx dec) I \det dx 



dy, dx\ 

 'TJ' 



In order to deduce the true expression for — — ,we mustdecompose,£uccessively, 



dx 



the triple symbols of substitution into simple ones, and afterwards replace the sym- 



flv fi~ el ? 

 bolical derived functions — , — -, — , by the real ones which they represent, and 



dec dec dx 



which are determined by a prefixed symbol of substitution. 



This rule serves to effectuate with facility all the reductions which have to be 

 applied to the variation of an integral ; but to enter into further details at present 

 would be to demand too much from the patience of this illustrious assembly- 



I shall merely add a short remark relative to the application of the calculus of 

 variations to the investigation of the maxima and minima of definite integrals. 



In seeking the absolute maximum or minimum of any integral S, containing one 

 or more unknown functions, it is merely necessary to introduce into these functions 

 an arbitrary parameter* in order to reduce the problem to that of finding the maximum 

 or minimum of a given function. In fact, by so doing, the integral S becomes a 

 function of os, and its maximum or minimum is determined from the equation 



^ = 8S=0. 

 da 



But if, at the same time, certain other integrals S p S 2 , &c... are required to pre- 

 serve the constant values 



S t = Cj, S 2 = c 2 , &c 



during the variation of the unknown functions, the method just indicated does not 

 suffice. In this case, which notwithstanding its frequent occurrence has scarcely 

 ever yet been treated in a sufficiently rigorous manner, the result may be arrived at 

 by a very simple expedient. Into each of the unknown functions let a number of 

 parameters a, /3, y ... equal to the number of integrals S, S v S 2 ... be introduced; 

 these integrals, which thus become functions of the parameters, may then be repre- 

 sented by 



S=0(«,/3,y...), S 1 = yfr (*,fty...), S 2 = x («,fty,...). &c 



and it now remains to find, amongst all the values of cc, /3, y which render 



S 1 = c 1 , S 2 = c 2 , those which correspond to a maximum or minimum of S. Ac- 

 cording to the known principles of the differential calculus, it will suffice, therefore, 

 to find the absolute maximum or minimum of the sum 



S + aS^JS,-!- , 



