84 
MR. J. H. JEANS ON THE EQUILIBRIUM 
We have also =■ 0 whenever m is greater than n. Hence we see that the 
determinant on the left hand of (54) reduces to the products of the terms in its 
leading diagonal, and that the ecpiation itself is equivalent to the separate equations 
— or 1-277 p) .(62) 
taken for all values of n from I to co . 
Corresponding to a i“Oot of (62) there is a point of bifurcation, and the linear series 
starting from the point must be found from equations (52). From these equations it 
appears that the linear series coiTesponding to a root of (62) will be such that, as far 
as the first order of small quantities, exists only for the values s = 2, n, n — 2, 
n — 4, . . . . 
Of these series the series n — 2 may be rejected, as corresponding only to a step 
along the series of elliptic cylinders, and not to a new series at all, and the series 
n = I may be rejected, as corresponding merely to a change of origin. 
We are left with the values n = 3, 4, 5, ... , and for any one of these values we 
have, from equation (61), 
Pfn _ 1 1 + «“ 
da,I n 1 — 2a02 
The points of bifurcation are accordingly given by the equations 
1 1 + g" _ ^ 
n 1 — Safq 'lirp 
where n has the values 3, 4, 5, . . . . 
Tliese points of bifurcation are points on the series of elliptic cylinders, hence w'-, 
Uo, and a are connected by equations (57) and (58). If we eliminate or and fi’om 
the three equations (57), (58), and (63), we find, as the equation giving iDoints of 
bifurcation of order n. 
1 - 1 + 
O 
( 64 ). 
This equation must be solved by graphical methods. In fig. 2 the curve which is 
concave to the axis of a is the parabola 
y = i(l — .(65). 
