STABILITY OF THE PEAR-SHAPED FIGURE OF EQUILIBRIUM. 
13 
0 X , we must put cos 2 6 l = l — %, so that q 2 = k 2 [1 + ( — cos 2 0 X )]. Similarly for the 
K~ 
harmonics of rank 4, two roots correspond with imaginary angles, and so forth. 
Subject to this explanation we may now regard the roots as defined by 0 U 0 2 ...0 n . 
Since A r 2 = 1 — q 2 sin 2 y, we have A 2 = I — k 2 sin 2 y sin 2 $ x — 1 — sin 2 (3 sin 2 0 X , and 
{3 2 n 2t . (/*) = cosec 2 ” y ApA/... A, 2 . 
If we write 
D x — (sin 2 0 X — sin 2 0 X ) 2 (sin 2 0 X — sin 2 0 2 ) 2 ... (sin 2 6 X — sin 2 0 n ) 2 ( n — 1 factors), 
our former C x may be written in the form K in ~ i D x , and the several coefficients in the 
expression for may be expressed as trigonometrical functions—some of which may, 
however, he hyperbolic. 
We thus have 
a*. 21 = -Wj 11 0 - si “ ? 0 si & 2 W i F M s 
:k i 
1 
i D r sin 2 0 X cos 2 0 r 
n i 
^ f D c sin 2 0 X cos 2 0 x (1—k 2 sin 2 0 X ) 
+ K 2 sin y cos y cos /3 % 
1 
D x cos 2 0 X (i — k 2 sin 2 0 X ) (1 — sin 2 /3 sin 2 0 X )_ 
This formula agrees with the result given for <H 2 S (s = 0, 2) in § 4 of “ The Pear- 
shaped Figure,” although the formula is there expressed in terms of q 2 , and 1/(k 2 — q 2 ) 
is replaced by its equivalent (1—2 q 2 )/q 2 q' 2 . 
In the case of the even zonal harmonics of order i, all the 0’s are real angles, and it 
facilitates the solution of the equation for 0 to note that, with rough approximation 
(improving as the order of harmonic increases), 
0 1 
n OH 
03 = Yi 
( i-l)n 
2 i 
The following numerical values apply to the critical Jacobian ellipsoid :— 
For the fourth harmonic 0 X = 20° 15', 0 2 = 61 c 11'; the rough approximation gives 
22° 30' and 67° 30'. 
For the sixth 0 X = 14° l /- 9, 0 2 = 42° 12 /, 2, 0 3 = 71° 8' - G ; the rough approximation 
being 15°, 45°, 75°. 
For the eighth 0 X = 10° 43'T, 0 S = 32° ll'*8, 0 3 = 53° 51'"2, 0 4 = 76° 21'*8 ; the 
approximation being 11° 45', 34° 15', 56° 45', 79° 15'. 
For the tenth 0 X = 8° 40', 0 2 = 26° 1', 0 S = 43° 26', 0 i = 61° 3', 0 b = 79° 28'; the 
approximation being 9°, 27°, 45°, 63°, 81°. 
The values of the several <H’s found by quadratures were in every case too small: 
