STABILITY OF THE PEAR-SHAPED FIGURE OF EQUILIBRIUM. 
8 
§ 1 . The Rigorous Expression for the Ellipsoidal Harmonic Functions. 
Rigorous forms have been found in previous papers for all the harmonics of orders 
up to the third inclusive. For harmonics of the fourth order rigorous algebraic forms 
may be obtained in all cases except when s = 0, 2, 4, but these are exactly the cases 
to be considered in this investigation. We have, then, to show how these functions of 
higher orders may be evaluated rigorously for an ellipsoid of known ellipticity. 
The only case required is that in which both i and s are even, and although all the 
forms might be evaluated by processes similar to those indicated below, I shall confine 
myself to this case. 
We have seen in “ Harmonics” that if /3 denotes (1 — k 2 )/(1 + k 2 ) of this paper, and p 
denotes sin 9, 
W (p) = Pi W-P<M*Pi + * (p )+ P 2 q s+i Pi +i M - ) w ~ s) ^%Pi (p) 
—figs-zPi " (p) + frq s -iPi i (p) — •••+ (-)~ s fi- s qoPi (p)- 
It is well known that 
and 
l i \ dp 
P s ( ll) _ {i+s)(i+s-l)...(i-s+l) , 8 u.f i-, (z-s)(i-s-l) i-,- 3 /, 2 x 
’ {l ) W77! V 1 h) P i! 2 2 (.s + 1) 1 1 M 
(•s+1) 
I )Ji-s-2)<fi- s - _3) i -.- 4(1 _ 2)2 _ 
2 ! 2 4 (s+ 1 ) (s + 2 ) 1 1 H) 
Hence we may clearly write in the form 
Pi 5 (p) = f 0 sin 1 9—f sin 4-2 0 cos 2 6 +f sin 4-4 9 cos 4 9—.... 
Since when 6 - is not zero P/( 1 ) = 0 , and when 6 ' is zero P t (l) = 1 , it follows that 
f = ( — y- s /3- s q 0 . For the zonal harmonics (s = 0 ) this gives f 0 = 1 . The determination 
of the other f 's depends on that of the q s, which we shall consider later. 
Another form of p/ (g) will be useful, viz. : 
W {p) — ci~b cos 2 9+c cos 4 9—d cos 6 9+e cos 8 d— .... 
It is obvious that 
a =/o, 
b=A + dr\ f - 
c — f i r , i r 
Ji 2 . 1! ^ 2 + 2 2 .2! ^ 0 ’ 
d — f 4- f 4 - Q-~2) (ft —4 ) r i (^—2) (t — 4) r 
&c., &c., &c. 
Thus, when the f’s are computed it is easy to obtain the a , b, c, d, &c. 
b 2 
