OF EQUILIBRIUM OF A ROTATINC4 MASS OF LIQUID. 
321 
1st. The equation 
W Qi^ (^) ”■ P*' (^) (^) 01’ {^) Qi" (i^) = 0, {i > 2) 
is not satisfied by any value of v between 1 and infinity, if or P/ is divisible by 
1 + 
O 
- 
It appears from the forms of the functions as given in § 4 of 
“ Harmonics ” that the P functions are so divisiljle. These functions appertain to 
the types EES, OOC, OES, EOC, and therefore the ellipsoid cannot liifurcate into 
deformations of these types. 
2nd. The equation has no solution if p," is divisilde by (w— l)h We again see 
from § 4 of “Harmonics” that is so divisible if it is of tlie types OOS, EOS. 
Hence the ellipsoid cannot bifurcate into these types. The only ty])es remaining 
are EEC, OEC. 
3rd. The equation has no solution if any of the roots of ^1/ (v) = 0 lie outside tlie 
limits + 1 to — 1. The only of the types EEC, OEC which has all its roots inside 
the limits + 1 to — 1 is the zonal harmonic for which s = 0. 
Hence the ellipsoid can only bifurcate into a zonal harmonic. 
4th. The equation 
Pi'Qi'-l^ia = o {i>2) 
must have a solution between 1 and infinity for all values of i 
It follows from these four propositions that the Jacobian ellipsoid is .stable for all 
deformations except the zonal ones, and that as it lengthens it must at successive 
stages bifurcate into each and all the zonal deformations. 
5th. As the ellipsoid lengthen,s, the first coefficient of stability to vanish is that 
of the third zonal harmonic. This stage is the end of the stability of the Jacobian 
ellipsoids, and there is almost certainly exchange of stability vuth the pear-shaped 
figure defined by this hai-monic. 
6th. It has not been rigoroirsiy proved that there is only one solution of tlie 
equation = 0 even in the case where i = 3, but M. Poixgaue believes that tliis 
is almost certainly the case. 
7th. The functions 
PJ (^o)] 
or > 
PJK)J 
X 
or > 
P/ (^o). 
have always the same sign as increases from pq to infinity, provided that .v and t 
are both greater than zero, and i greater than 2. 
The seventh of the preceding propositions renders it easy to determine the relative 
magnitudes of all the 3^’s belonging to a single degree i. 
In what follows I may take the symbols including also P, Q. 
VOL. CXCVIII.—A. 2 T 
