Combining the pairs of conjugate partial fractions according

to equation (29), the first pair becomes

and similarly will the other pairs of conjugate partial fractions be compounded; so that the following series will be formed,

If n is even, then equation (48), by means of Art. 64, Vol. I,

becomes

1

=

x2-2x cos

+1

Let n be even; then, in Art. 65, Vol. I, it is proved that the

roots of a*+1 are cos±√lsin,

3п

3п

COS +V-1sin ,

n

n

roots of "+1 = 0; therefore the coefficient

and combining the pairs of conjugate partial fractions, according to equation (29), the first pair is

and the other pairs give similar results; so that

each of which must be integrated according to the process in

dicated in the preceding article.

Again, let n be odd: then the roots of "+1 = 0 are

so that if the conjugate partial fractions are combined by a process similar to that employed when n is even, the last pair becomes

п

2 4

1 3п

-colog (r-2.rcos+1) + sin tan-1

1

4

4

4

2 4

3п

x2-2x cos +1

5