of a Function of ny mimber of Variables. 419 
Therefore, 
du du du 
(8.) 1Du=D (du~7, ot = ay ty - 5 dz -&e.] 
dDu dDu d Du 
oo a OES dy ae wa Pp dz+ Ke. 
which is the same as the equation (3) before obtained. 
If Du be put for wu in the equation (4), then 
(g.} 6Du=1Du4+rDu; 
and if this be compared with the equation (5), it follows that 
(10. 4Du=D4u; 
which is of the same import with the above equation (2). The 
equation (8) may be presented in another simple form, that is 
worthy of notice, when J+n+m+&c. =1; as for example, 
when 7=1, and 0=m=n=«e. for then 
du d du du du 
Vide de tat oy a ye bz &e.) 
d?u ‘ d*u d?u a 
taeda © dedy YT ded 9? + 
or 
Py du du dudde duddy dudiz 
SEER 0 ee ey det AE Ee es 
Which is the same as the former equation (1). bape 
Mr, Delaunay, in a paper on the calculus of variations lately 
aerate and which has particular reference to the variations 0 
Multiple Integrals, has thought it sufficient to vary the limits of 
Integration, and the form of the function which the dependent 
variable is of the independent variables, without assigning to the 
latter any variations; in which case no investigation is needed to 
determine the variation of a differential coefficient of the depen- 
dent variable ; for it is obvious at once, tha 
gitar &e.94 ar &c. 
Py 
“da'dy"dz" &ec. du'dy"dz" &e. \ 
Now With reference to the variation of a double definite inte- 
Sral, it is plain that if the integral be presented in the following 
A OF | — 
form, f. dx J. V, and be considered a substitute for such an 
— (2) 
Smpreilot:ts this, viz, WW one sy gay Wyn oy 
cHA z=4 
‘ dw dW dy 
where W is such a function of «and y, that F Ty ae U; 
and U such that “=v, then the method of Delaunay will suf- 
