12 
SIR G. H. DARWIN: FURTHER CONSIDERATION OF THE 
III all the types 
B r = 2 
fix-(-lx 
+ 
lx-qi 
+ ... + 
<1 
with the omission of the term which would be infinite. 
(J 
We have generally B = -p|i (v 0 ). Hence by logarithmic differentiation of 
H wo) dv 0 
the expressions for the several types of ^ we find results given in the following 
table :— 
Table of the B Integrals. 
Type. 
i 
order of harmonic. 
s 
rank of harmonic. 
Bd 4- 2 sin 
EEC 
2 n 
2 1 
Il/A* 2 
i 
EES 
2n + 2 
21 
b sec 2 (3 + b sec 2 y + -1/A, 2 
i 
OOC 
2n + 1 
2t+ 1 
}sec 2 7 + 21/V 
i 
OOS 
2n+ 1 
2/+1 
\ sec 2 /3 + 21/A^ 2 
i 
OEC 
2n+l 
2 t 
-1 + 21/A., 2 
i 
OES 
2n + 3 
2 1 
b + b sec' 2 fi + 1 sec 2 y + 21 /A x 2 
1 
EOC 
2 n + 2 
2/ + 1 
.i_ + i sec 2 y + ll/A a 2 
i 
EOS 
2 n + 2 
2/+1 
b + b sec 2 /3 + 21/A, 2 
i 
In the case of the zonal harmonics (s = 0), q j: /K is always less than unity; for 
harmonics of rank 2 one of the q^K is greater than unity and the rest are less ; for 
rank 4 two of them are greater than unity and the rest less. 
2 
For the zonal harmonics there is some gain in simplicity by putting sin 1 ’ 6 X — 
K 
We then take the equation 
(/x) = a—b cos 3 0+c cos 4 6 — ... — 0, 
and find all the n roots, say 0 X , 0 2 ... 6 n . 
If we solve the corresponding equation |3 2 « 2 (/x) = 0 for the tesseral harmonic of 
rank 2, we find one root for cos 2 9 to be negative. If this root corresponds to 
