172 
MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
n TT 
LG 
tx, q 
— cos0cos(Bcosfl + ® i. 
S* COS 2 ^cos ‘ 
cos^^ cos 0 cos <p sin (tan ^+tan <p)^ 
^^|^^cos-_|_g-cosjjcos sin^|{£'*'°®® COS (|M/ sin ^)} 
Hence we find, by comparing (A.) with (B.), 
^77 r cos 29 sin 25 ^ & 
\ c?^£'"“"®cos (jO* sin ^)|2s cos sin 
We liave already proved that 
in ||=T. 
. , 2 , 2.3 a:* , 2.3.4 3^ , , 
1 + j^+t- 7 •— +:i-^ +&C. 
4.5 1.2^ 4.5.6 1.2.3 
= :;5 (-^ + 2) + 73 — 2) s*. 
Hence 
i_j_! ? . ^ „ 
^■^"5 2 ^" 5.6 2 . 3 ~^^^‘ 
12 / , . 12 , , 2 
= ^(jo+2)+-,{x-2y--, 
. (B.) 
and 
2/y,2 »/3/r3 
1 +^-^+ y:^+ +&C. = g'^. 
Consequently the theorem of M. Smaasen will give us the sum of the series 
3 ju-a? I 3-4 , 3.4.5 ^ 
^ ”^2’5.l"'~^'^6.I.2"'~ 2.3.4’5.6.7.I.2.3"i"^^‘ 
by means of a single integral, and we obtain 
TT TT 
f f' d6d(p 
g2^cos0cos(5cosC0 <p) cos®^cos^<p COS { 2/x; cos ^ COS ip sin (^— ^)+ tan (p— 5^} 
^ c?^{6(cos 3(?4-2 cos 4^) + 6£'°®® cos (3^— sin^) 
..’0 
— 12£“®® cos {40— sin 0 ) — cos cos {[h sin 0). 
The fundamental idea of the preceding calculations, as will be readily seen, is as fol- 
lows : to reduce every term of the series proposed to be summed by means of definite 
integrals to the form of the general term of the series whose sum is given by the 
common exponential theorem, and then to find the sum of the whole quantity con- 
tained under the signs of integration by means of that theorem. The factorials in 
the numerator of each term may be taken in any order we please relative to those 
of the denominator, provided that the same relative order is observed in every term 
throughout the whole series ; moreover, we may use different integrals to express the 
