EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
427 
If the energy of a system be expressed as the sum of a number of coefficients, each multiplied by tbc 
square of a parameter, it has been shown by M. PoiNCARh that the stability of the system depends on the 
signs of these coefficients, which he calls “ the coefficients of stability.” But, if the expression for the 
energy involves the products as well as the squares of the parameters, the coefficients of stability are the 
roots of a determinantal equation involving the second differentials of the energy with respect to the 
parameters. 
Let V, a linear quadratic function of x, y, z, Ac., be the energy of a system in equilibrium; then the 
determinantal equation is 
d-V . 
dW 
d~V 
Ac. 
dx dy 
dx dz ’ 
dW 
dW V 
dW 
Ac. 
dy dx ’ 
dy* ’ 
dy dz 
dW 
&V 
<^-x, 
Ac. 
dz d x 
dz dy ’ 
dz 2 
Ac., Ac., Ac. 
The solution of this equation in X involves the determination of the several fundamental modes of 
vibration of the system ; and the roots are the coefficients of stability. 
Now suppose that V involves a constant, then, in causing that constant to vary continuously, we have a 
series of systems of equilibrium of the same kind; and the coefficients of stability vary continuously at 
the same time. If the system be initially in stable equilibrium, the stability ceases when a coefficient of 
equilibrium vanishes. The system at the moment of instability is in a condition of “ bifurcation,” that 
is to say, there is another seines of shapes of a different kind, of which this shape is a member. In 
making the constant vary past the critical value, we find this second series of shapes stable, whilst the 
first is unstable. 
If the system be in uniform rotation, so that instead of absolute equilibrium there is equilibrium 
relatively to uniformly rotating axes, the same theorems hold true, provided that only one root of the 
determinantal equation vanishes at a time. 
This Iqst is the case which we are considering, and the constant, which we suppose to vary continuously, 
is c, the distance between the two centres of the mean spheres of radii a and A. 
When the two masses are far apart the equilibrium is stable, but when they are brought closer a time 
may come when one of the coefficients of stability vanishes. 
The condition for the vanishing of a coefficient of stability is determined by the determinant (x.) with 
X= 0. 
To find the determinant, we have to evaluate the second differentials of E with respect to the 
parameters n, N, p, P. 
If we form the determinant corresponding to (x.) with X = 0, it is obvious that two infinite squares 
of entries which are diagonally opposite to one another, and which meet at a corner, are to be filled in 
with differentials involving dndp, dn dP, dN dP, dN dp, in the denominators. All these entries are 
zero, and hence the infinite determinant splits into two independent infinite determinants, one only 
involving the differentials with respect to N, n, and the other only those with respect to P, p. The N, n 
determinant may be called “the tidal determinant,” the P, p one “the rotational determinant”; for the 
origin of the terms in each is obvious. 
By considering only the tidal determinant, we see how the other may be treated very shortly. 
For the sake of brevity, write 
Wk Ni — 
d^E/dn k dNi 
(- PE/dru?)i (- d 2 E/dNr')i 
(xi.) 
Then stability vanishes, as far as regards the tidal forces, when 
3 i 2 
