792 PEOFESSOR BOOLE ON THE COIVIPAEISON OF TEANSC'EXHENTS, TTITH 
stituting and observing that the limits of are — cc and cc , we have 
T =J . . dy,dy., . .dy^co&\J^a— Ifs — 2 [syl + l/i, “ ^ ^ j 
. . dy^dy ^ . . dy^ cos | (^—-h^s — '^+ 2i’Sj/^+ y|. 
Let 
As 
( 3 .) 
Then ^=1 ..dy^dy^..dynQ,o%\{a—(r)v-\-^vsyl-\-i'^—-^^ 
{Vsf 
COS Ua — 0-)t’ + ?^— ??7L 
whence 
V= 
where 
r(d 
^00^00 
JpJo Jo 
s' i-‘ cos |(«-<r>+ = ds s * ^Q. 
Q=:;^|' ^ dadv v^~i cos Ua~-G)v-\- 
Pqja 
= (~^)' <ia!i»cos{(a-<7)»}/(c!) 
=(-^)"Vw 
by Fourier’s theorem, f (a) vanishing when o- does not fall within the limits ^ and q. 
Thus, finally. 
V: 
— r 
Twjo 
ds s ^ * 
7 . n 
a \ «-2 n, \ 
■*; /W’ 
0.) 
the fully expressed value of s being 
(T=- 
We may remark, that as 
(T — fi- 
a increases with s. As s varies from 0 to oo, o- varies from — cxi to oo. Thus, whatever 
may be the values of p and q within which the variation of g is confined, there will exist 
corresponding positive values within which the variation of s null be confined, and those 
limits must take the place of 0 and oo in the expression of (4.). 
39. In strictness there is no need of referring to the limit in the statement of general 
theorems like the above involving an arbitrary symbol of functionality. The consistent 
interpretation of that symbol will suffice. Thus tlie results at which we have arrived 
are virtually included in the theorem 
i ..dx^dx^ 
I — oo 
, dx 
y(/].X'j -)- • -f- L^K.) 
" { 4 ^ + («1 — + («2 — ® • ' + ( a « — 
n ^0 
TT^ [ 
ds. 
(6.) 
G having the value given above. For if l^x^-\-l. 2 X.^ •• -\-lp'n is confined witlim given limits, 
./*(^i^i 4-4^2 — Vh^n) niay be regarded as vanishing whenever l^x^-\-l^x^-- -\-l„x„ transcends 
