stability of rotating masses of liquid 199 



which there belongs a function Hq, then a contiguous form whose 

 equilibrium is under examination, to which belongs the function 

 H, and then, further, a "virtual" form, which has the same angular 

 momentum as arises in H, to which the function H' applies. We 

 may then suppose the parameters, above spoken of as identifying 

 the form, to vanish for the form {Hq), may denote their values for 

 {H) by X, y, ..., and their values for (//') hy x + ^, y + rj, .... We 

 put ^[x^ — ^fjLQ^ = k. Presuming certain conditions of continuity 

 for the functions involved, the equations of equilibrium of the 

 form (H), which are such as dH/dx = 0, dH/dy = 0, ..., must, for 

 X = 0, y = 0, ..., together with ^ = 0, be satisfied for (^o)- ^^ 

 general (certain conditions being introduced in the choice of the 

 parameters) these equations determine a form (H) corresponding 

 to every arbitrary small k; the necessary and sufficient condition 

 however that (Hq) should be a so-called form of bifurcation, or 

 branch form, is that these equations should lead to more than one 

 form (H) for any given small k. A sufficient condition for the 

 stability of the form (H), is that the quadratic form, in the arbitrary 

 variables f, 17, ..., consisting of such terms as 



^ed^Hjdx^ + ir^dm/dxdy + ..., 

 should be definite, and be positive, when, therein, the coefficients 

 d^Hjdx^, d^H/dxdy, ..., are those functions of k arising by sub- 

 stituting the value of x, y, ..., just found from dH/dx ^0, 

 dHjdy = 0, etc. And, in particular, if {Hq) be stable, this quad- 

 ratic form must be definite and positive for /^ = 0, a; = 0, ^ = 0, .... 

 Conversely this last fact, when (^q) is known to be stable, con- 

 siderably reduces the labour of considering the stability of {H). 



§ 2. In our case, {Hq) is an ellipsoid, and {H) a contiguous, so- 

 called pear-shaped, form. Sir George Darwin calculates {Papers, 

 III, 349), a form for the increment of the Lagrangian function 

 W + liJi^lI, in passing from the form (H) to the form {H'), which 

 leads, to the same approximation, to a form for SH, or H'— H, 

 which may be written thus: 



3iP^2yto = ^'(^-^^'-^^^^-^'^ 



+ lax^ + ^by^ + ^cz^ + gxh + hx^y + fyz 



+ i:[h'xY + Wy'^V, 



for facility of comparison we may give the equations which connect 

 the notation here employed with that used by Sir George Darwin: 



X, y, z, y' ; ¥, I, m, n, 



are used respectively in place of 



e L fo('-> /;(«) • - - - _ - • 



