1882.] Liquid Ellipsoid about an Axis. 210 



The conditions for steady motion become 



<o l <w 2 eo 2 fJ,' 



g_h_k_ _E 

 p q r fi ' 



, (/ W k' 



and -, = — = — = — /*. 



p q r 



where E = &V + cV + a 2 b 2 - 3a 4 ; 



also g, p, g', p are zero. 



6 2 c 2 



Putting as before -5 = x, —*—y\ 



then 



a 



(y 4 z) M 



(x + z)N 



3 {x-yfH> 



where 



M = x (x -y)A-x(4<- Sx - y) B + y (1 - x) (4 + x - y) C, 

 JV= - y (x - y) A +• x (1 - y) (4 - # + y) B - y (4 - x - 3y) C, 

 H = xy — x —y + 4, 



and s = 1 + 4 , 



a 



so that 2 is given by the quadratic 



z 2 — (4 — x — y) z 4- xy = 0. 



Therefore from a; = to x = 1 (region I.) ZT is positive ; from 

 x = 1 to u? = 4 (region II. A) H is positive ; from x= 4 up to the 

 curve 11=0 (a hyperbola) i> is negative; and beyond the curve 

 II = 0, II is positive. 



In regions I. and II. A, y + z is positive ; in regions II. B (that 

 is, from x = 4 to x—<x, and y = to y = l) and III. y + z is 

 negative. 



In all the regions x + z is always positive. 



Also writing M in the form 



M = 27rp(l-x)(x-y)j^ |0/-4)^ + rr y -x + 7 / -4J- /w _ , 



16—2 



