Mr. Boot on a certain Multiple Definite Integral. 143 
Integrating by this formula the second member of (7), 
n 
7? 
ashe -cos(T) = 5 = +h, 2 = Bao (w—(—*, 5 - «jew 4. 7 
and substituting in (6), 
wo 
v= me : ‘ dadvdw —— wy eae cos (w (ss apt ° .)e 
2 zs) : (a) = 
— Nie 
mn )f (a). (9) 
v v ‘c ; 
Wet —— athena ——— pe and the limits of s are 0 and ~, whence, 
s 2 
we —1 
a, ) : 
mY “dad i@aiae a) pak ma (ao —(aat a 
are 5 
-_ * 
hih...x IR a? 2i-— . 
= apd AN Ne dadeds a att Si cos( ao— i ‘op Aaa) s (a) 
1 1 eS 2 i—— 
Wilson ( inet 2i— 7m \-. 
= 1G) ~o0ve ds lr Brae cos ((a--) lar = \F (a). (10) 
Sa” Q 
n 
if, for simplicity, we write 
g= (sth) (sthy)..(s thy) | 
tee 
a,” a,” an 
et aie a 
Z mv =(—) = cos(a—a)v. 
Substituting this value in (10), we may express the result in the following form, 
SiGe +3 (iGsNtaa eit 
Ri sca Ag ) ) Ms 
as = Sr | 
T(?) hi ore a ob 
: iM Rate 3 
wherein Qg = ay \ dadvcos(a—o)vf(a). But by Fourier’s theorem this value 
WiueetO ven O) 
Hla 
Now cos( 
of Q is equivalent to f(o), when o lies between 0 and 1, and is equal to 0 when 
« transcends those limits. Hence, substituting for Q, we finally get 
