MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
177 
Again, we have 
1 Q4m+l oim+l 
1 — &c —0 
- 27 rn^ R tt V/, 
pir 1?”^ p27r_p— 27r I Stt 
We must transform the element thus : 
1 — -I / l\4m ^1 logej; ] / j \ 4 j« 
Again, 
Lastly, 
logeJ^ 
j » : 
v' n r“ 
2v'7r logsa;J_ 
7 ?z 2 
2 
Hence, combining these integrals together, and substituting for 
as before, 
we are able to transform the above series into one which can be summed by the ordi- 
nary rules. The resulting definite integral will of course be equal to zero. 
Cauchy has applied the methods of the residual calculus to the determination of 
the sum of the series whose general term is 
-Tia e— Tta 
.TlTT c 
n cos ncx, 
+ 
1 
in finite terms. We may transform the element - 4 -— 3 <^hus : 
Tt C 
1 
+ 
sin &zdz. 
Again, 
2 
*00 
£ £ *"cos2n2. 
V TiZ 
Wherefore, combining these integrals, and transforming the other elements as 
'ft 
before, we may find its sum by means of definite integrals. We may resolve 
into its partial fractions, and then find the sum of the series, which would be 
simpler. 
The transformation of which I have used above, is due to Professor Kummer, 
who has applied it in the seventeenth volume of Crelle’s Journal, in a paper to 
which I am indebted for many ideas relative to the connexion of definite integrals 
with series, to the expression of the series 
and others of a similar nature by means of a definite integral. The integral 
J fil'st applied to the summation of series, whose terms involve elements 
of the form by Poisson in his Memoir on the Distribution of Electricity in two 
electrized spheres, which mutually act upon each other. He proves that the cal- 
