of a Function of any number of Variables. 421 
Therefore, 
dV du du du 
(13.) aontege D(tu— i fs nay rae ee 
dV dV . dV 
Eee Oa + oy Uy ie dz+Kce., 
av av" ev : 
where P= Du, and ——, dy? dz? &c., are complete or total difler- 
ential coefficients with respect to z, y, z, &c. 
One other theorem, which has not, we believe, been elsewhere 
stated, seems worthy of notice here 
t V be a function of several variables, £,Y, z, &c., either 
Yieiale or constant in its form with respect to them; then by 
virtue of the omni (1), we readily obtain the following, 
dV dV 
da I +dy 97 of. iy as Ps zy +k. 
w dV _ av 
dV dV 
Now seit dV asa uniform function a ry a qe &e 
dV 60 dV 
de, dy, dz, &c., viz., the function ie + Gy dy Sine es dz+&c., 
we have the equation, 
Ad il 
. ddV=dr oP le ig * Oe fs +&c. 
(15.) 
dV 
+ ate ys iy Y ty +o ddz +&c., 
which one with the last gives the lowing 
dV 
daV—asy 4. Te (dde-ildz) +5, ™ (sdy-<dby + F- ——(ddz — ddz)+&c. 
and if we put the esac 6 for aa as) 
(16.) Va Oat hy ha 6z+&e. 
which is the theorem that was to be stated. 
If either of the variables x, y, z, &c-, as for example 2, is a func- 
tion of the rest, span ro in its form or not, then 
a Ora ty + Ge 6z+&c. "y 
dV de av), , (dV de, dV) av dv 
“(Zz ag ay) ae gat da) tee tay ae’ © 
