PiiOFESSOK G. 11. DAPWIN OX ELLIPSOIDAL HAILMOXIC AXALVSIS. 
oy.) 
The integral of an odd power of cosO i.s easily deterniined, and it will be fnind that 
the result (86) is obtained. It is, however, clear tliat if n is not greater than 2 the 
development in powers of sec’ 6 is not legitimate. 
When n is not greater tlian 2 the formula (84) of the last section is necessary, and 
^\■e find 
1"-^ ‘.w = ''^p- log 
(«= - (/3+^/3-') log l+y ( 1) r 1 +■>+ p 
{d0 
A 
Ki+i^+),y)ic>gy-h^' 
. . .(S7). 
We have now' to find the second integral E, and this may be tlone more easily than 
by reference to the general formula of the last section. 
We have 
E = COS'" $AcW ={1 — K' 
fcos-"+- 0 
cie-^K^ 
j'cos-'* 0 
dd. 
de+- 2 fi (1 -^) M 
It wall be observed that eveJi when n is 2 the general formula (86) gives the 
I) integral as far as the first power of y8. Hence in finding K we may use that 
general formula except when n — 0, 1. 
2/7+1 
Then since/'('O ——-/(».+ !)yHien n is greater than I, 
K=y(a+I) 
=/('•+') 
(3 (4/i + 5) /y , 2// + J 
_ „-./3+2/3p,+ + ^ 
2// Sudn—ly 
( 88 ). 
But when h = 1, 
E = (l -2/3 + 11/3^). i/r log| -2/3(1-13}. 0(l+m log'l 
+,/'C->l [(I -2/3 + 2I3-) (I - II3-UI3’} +3/3(1 -13) {1 + i^)J. 
= —1/3’ l^'g'"^ +,/ (2) -h+/3+ 1. 
And when n = 0. 
E = — (I — 2/3+-/S;)^(l+'f/d) +-/3 (I — ft) (I -hi/S) log -- 
+./ (') L(' ~'-^ftft-'^ft') (+ ''iftdrTift') ~'-^ft {^~ft) ■ iv/^' I’ 
—ft{^~^ft) lt>g ftrf (1) [1 — i/3+ . 
(89). 
(90). 
