178 
MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
culation of the electrical arrangement depends upon the value of the definite inte- 
gral, 
sin cz.dz 
— 1) [a -\- b cz) 
I mention this on account of its analogy with the definite integral 
dz sin 2’zz 
Jo (e"" + 2e"cos 2 t0 + 1)2 (/^"^-I)’ 
whose value is found above. The principles contained in this paper will enable us 
at once to find the sums of the series 
1+^-1 
H 
oc^ 
Ws 
X* 
1.2 ' 1.2.3^ 1.2. 3. 4 
tan fl . tan 2fl . tan 3fl 
H-&C. 
1 
1.2 ' 1.2.3 
■&C. 
„ sec- 
sec . 2 
sec- 
1.2 ' 1.2.3 
■&C. 
and of many others which can be imagined, by means of definite integrals. The 
definite integral of Poisson given above occurs in the solution of a functional equa- 
tion ; and it is probable that series similar to those I have been discussing in this 
paper, may be useful in enabling us to express the solutions of other functional equa- 
tions by definite integrals. 
