420 On the Variation of a Differential Coefficient 
fice, and the variation of the integral may be exhibited with 
entire generality in the following manner, : 
"A (x) “A F(x) A 
8 fu dy va fix fay OV+ | dz ( Var(x) —Vaf(a)) 
av f(x) aw f@) a y= Fix) y=fiz) 
‘ . we) . af ts. y 
(12.) +34f 4, Vioa7 taf Vewei 
where 8V denotes the variation in the form of V considered as 
a function of x and y. But it may at times be necessary or con- 
venient to regard a double or other multiple definite integral as 
the sum of an infinite number of infinitesimal elements, or more 
correctl rhaps as the limit of this sum, and variations may 
need to be attributed to these infinitesimal elements severally 
rather than in the aggregate, in which cases, (as has been re- 
marked by Ampére in a paper on the Calculus of Variations ap- 
plied to mechanics, ) it is not sufficient to regard the variations of 
the independent variables as nothing, but general values m 
be or them as well as for the variations of the de- 
pendent variables. 'The method of Delaunay is therefore essen 
tially deficient in generality. 
By means of the formule already stated, it is easy to obtain 
the proper expression for the variation of any function of both de- 
pendent and independent variables. Let V be such a function, 
whose variation is to be determined ; let the independent variables 
be z, y, Z, &c., and let P be one of the dependent variables. Then 
as V is a function of z, y, z, &c., and of quantities which are 
functions of z, y, z, &c., it is virtually a function of these inde- 
tion in respect to all the variables, dependent and independent 
variables 
Then in the general equation 6V=aV-+1rV, 4V must be the 
sum of all the partial variations of V, that are analogous to the 
I dV : dV 
following, 75 4P; which sum may be denoted thus > 7p 4P. 
But P may be a differential coefficient of a function: we will 
m+n &e. 9p 
suppose it such, and put instead of it, Du, that is, inayrde &e- 
Then, 
dV dV 
4AV=2 7p 4Du==7p Day sess D(su—Tu). 
