10 
SIR GL H. DARWIN: FURTHER CONSIDERATION OF THE 
In terms of xp 
1 
[f MT 
1 sin 2 xp 1 
7 
+ - tan 2 i/> + y sin 2 1 // + 2 
0 /i i A 
r sin 4 1 p B x sin 2 1 p 
XL V 
A t 
This lias to be divided by A and integrated, and the result will be expressible in 
terms of the elliptic integrals 
F(y) = 
E{y)-\ A dip, II (y, A*) 
•i o 
' 7 dip 
o A/A' 
Accordingly we require certain integrals, which are given on p. 313 of the “ Pear- 
shaped Figure,” but in somewhat different forms. Here and elsewhere k 2 denotes 
1 — k 2 , and q' x denotes 1 — <] 2 . 
The integrals needed are as follows :— 
,siu Wfy = _ i 
o 
A 3 
7 taw \p 
0 A 
7 sin 2 xp 
dip 
— - F (y) -i--— F (y) — sin . T . 9 o s . y 1 
K 2 [y> kV 2 [y> k' 2 COS P 
1 t? / \ , tan y cos ft [ 
—a (y) +- a — r 
K K 
0 
A 
d* = 4 F (r) - 4 E (y) 
^ = 
o A/A r 
7 sin 4 ip 7 , 
—i dip = 
0 A, 4 A r 
-4 it (y, A,)-ir(r) 
q/ q. 
2 \ 2 „ 2 ' .2 „ 2 
2g/\g7 2/ ' 7-g/) 11 (y ’ Al) + 2g/(g/ g?) F(r) 
i v / \ s ^ n y cos y cos /3 
”2 W + 2?7R-^)A/ 
( 1 ) 
The last two of these admit of considerable simplification, as will now be shown. 
It is proved in the “ Pear-shaped Figure ” that the n (y, A. r ) elliptic integral 
disappears from the expression for for all cases up to the third harmonics 
inclusive. I have also proved numerically that for the Jacobian ellipsoid the like is 
true for all the even zonal harmonics up to the tenth inclusive, and for the tesseral 
harmonics i = 4 s = 2 and i = 4, s = 4. It is thus certainly true in all cases used 
by me, and I do not care to spend perhaps much time in proving algebraically the 
general truth of the law. 
The last two of our integrals will occur in the form 
' 7 sin 4 xp 
o A/A 
d\p—B x 
’ 7 sin 2 xp 
o A/A 
dip, 
and I assume that the coefficient of n (y, A. r ) in this expression always vanishes, as 
is proved to be true in all cases actually computed. 
