UF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
327 
Ill order to determine the question as to whether or not it is possible that = 0 
should have another solution than that found in the next section, I liave computed 
the value of this coefficient for the Jacohian ellipsoid y = 75°, k = sin 81° 4'M, and 
find it to he —6 •627, From the manner in winch the numbers in the computation 
present themselves, it is obvious that for more elongated ellipsoids 3^3 will always 
remain negative, and will become numeiically greater. I have therefoi-e not thought 
it necessary to seek for an algebraic proof that there is no second root of the 
equation. 
Very long Jacohian ellipsoids tend to become figures of revolution, and the coeffi¬ 
cients of stability tend to assume the forms 
Pi (z^) Qi {v) 
IVWQd(v)' 
The forms of these functions are well known, and I think that fair approximations 
to the incidences of the successive figures of bifurcation might he derived from the 
vanishing of this expression. 
For example 
V — {v' — 1) loo 
+1 
V - 1 
Pr (^) Qi (^) = -ih 
{2,5v^ - 3(V H- 3) log 
z. + 1 
- 1 / 
- p (21z."- ll)(35z^^-30zv' + 3) 
I have not, however, attempted to solve the equation found by equating these two 
expressions to one another. 
Even when i = 3 and y = 69° 49' (the critical Jacobian) this rough approximation 
makes the coefficient of stability very small, hut it is to be admitted that and 
P 3 Q 3 differ very sensibly from {y) (i^) and ^3 [v) ©3 {v), although in such a 
way that the errors compensate one another. 
first term ve put y = 6 + -fK-sinScos 8 (as is clearly allowable in approximation) the term with coeffi¬ 
cient K- or sin -a disappears. This shows that it was necessary to proceed in the approximation as far as 
in order to obtain a residt. 
The methods of approximation adopted on pp. 326-7 are of doubtfnl propriety, but will, I think, lead to 
nearly correct results. There is, however, a mistake towards the bottom of p. 327 which runs on to the 
end. M. Kruger correctly points out that the second line of formula (24) p. 329 should run 
4 cos-a 
Q— log« cot (|- - U/). (-y- + 3 tan -y - 1 - tan -^y) - - V tan -y - tan 
4 ,, 
Sin y 
Lastly, on p. 335, line 13, for C = 0-3573, read C = 0-5379 ; and on p. 336, line 7, for 1-3573, read 
1-5379; and for - = 1-696, read- = 4-65. 
a a 
