CEETAIN APPLICATIONS TO THE THEOET OP DEFINITE INTEGEALS. 701 
Now V remaining unchanged, let w=vs. Then transforming in the usual way, we have 
dvdw='Vdvds, 
whence, on substitution, 
d(idvdsco^^a—x—ts)v-\-^^vs'~^f{a). . . . . . ( 6 .) 
In the application of this theorem to the reduction of multiple integrals, x and t will be 
replaced by functions of the variables involved in those integrals. Its advantages are 
the following. Like Foueiee’s theorem, from which it is derived, it enables us to ex- 
press any species of discontinuity in the function /‘(ir). Thus iif{x) is to vanish for all 
values of x which lie without the limits p and q, we have only to substitute p and q for 
— oo and oo in the mtegration relative to a. At the same time the theorem presents x 
and ^ in a functional connexion, which in all the most important cases renders possible 
the subsequent integrations without any new transformation. 
One subsidiary theorem remains to be noticed. Since we have 
^ d^cos{a±cf)='^cos(^a±jy 
we have, by successive applications of this theorem. 
(7.) 
( 1 -) 
r dij,dy^..dy, cos {a±^'z'lcyjl)=-^--^cos («+?) 
38. Example 1st. Let it be reqim-ed to evaluate the multiple definite integral 
■TT r 7 7 7 "P ^2^2 ' • P 
V = \ . dXMX, - • dx„ fTg , , \o , \2 ri \2Xi’ 
the integration extending to all values of the variables which satisfy the condition 
l\X-^-\-lpC2 • •~]rlnd'n 
If for convenience we represent l^x^-\-lpc 2 ..-\-lnXn by and {a^—Xyf-\-{a^—x^f.. 
+ («„ —x^"^ by we have by the theorem (6.), 
J A ^2^2 * * "P 
{/i^A (flj— {a^—x^'^.. + [an—0CnYY 
i] J] dadvdscos\^i—'2.lp(;^—{]d-^'2.a^—Xr 
The conditions relative to the limits will be fulfilled by introducing p and q for — co 
and oo in the above. Effecting this change and extending, as we then may do, the inte- 
grations relative to x^, x^..Xn from — oo to oo , we have 
j' ^ dadvdsd'd-~'^f{ajl:, 
where f cos 
Now lpc^-\-s{ar—xYf=s\Xr—a^-YfA + ^ if y,.=^r— Sub- 
