239 
and, as before, the complete evaluation of this symbolic form depends, in 
general, solely on the particular values of the given functions, F), F., fi, 
5. The results obtained in the preceding articles may be still further 
generalized. In fact, if the quantities a, B, y, &c., be supposed to be 
independent of the limits and of each other, it is evident that we may 
write the multiple definite integral 
ale ...- F,@F. YY) F: @ ... atyfst... dadydz.. 
Zz 
Yd a 
in the form 
F, (e2#) F, (22) F, (22) .. [| [Banyee . 2. dadydz ..., 
and then, solving the simpler integral, proceed to evaluate this result 
when operated upon by the symbolic factor to the left hand of the ex- 
pression just obtained. 
In the same manner, the multiple definite integral 
alain co (w — 1) Fs (y-1) Fy (2-1) ..a2y8ar.. de dydz.. 
wd Yid A 
is seen to be equivalent to 
r2(y 
F, (A,) F. (As) F(A ,)- a Ba ea ayer. . dedydz.., 
and the proposed question is reduced, as before, to the determination of 
the value of a simpler integral, and the deduction of the expression re- 
sulting from the operation upon this value, by the symbolic factor to the 
left hand of the formula last written down. 
6. Since, as is easily seen, 
eP=+Py | menY = mn. mn", 
we have, by successive operation, 
ePPaPy) | men! = (mn)? . mn. 
Hence, in general, if F be any algebraic function, 
F (eP2Pv) . mn! =F (mn) . mn’. 
By the aid of this theorem, we see that we are readily furnished with 
means for the reduction and simplification of certain other species of de- 
finite integrals. 
Thus the same suppositions as before being made with respect to a, B, 
it appears that the definite integral 
% | Y2 Xe [ye 
F oy de dy =¥ (ePatPp) PA bots 
Ea (ay) xe yP dx dy =F (e 3) ee vt y8 dx dy, 
