FIGURE OF EQUILIBRIUiNI OF A ROTATING MASS OF LIQUID 257 
It has already been remarked above that the squares of the semi-axes of J are 
_ 
sin- /3 ’ 
cos^ /3 
siir^/3 
I jf /3\ _ /d cos2 7 
1-/3/ sin- (3 
The mass of J is then ^-rrpk 
4 7 3 COS /3 cos 7 
sild /3 ■ 
I now take the mass M of the pear to be 
M = ^TTphZ 
cos j3 cos 7 
siiU ^ ' 
Thus Jcq is a constant which specifies the volume of lic[uid in the pear, and the mass 
of Jisif (yiyylo)-t 
It will be convenient to introduce certain new symbols, namely, 
= 1 — /c" sin^ 9 , F^' = 1 — k '^ cos^ 
= 1 — K® sin^ y sin~ 0, F^^ = cos~ y -{- k!^ sin^ y cos" 
where sin 0 is the /x of “ Harmonics.” 
The roots of the fundamental cubic were v~\ u", and -—and in the new 
1 — p 
. , t, . , « 1 — k'" cos- cb F- 
notation they are w, sm^ 9, -;;- or 
/C- K~ 
Since vZ = V-”, we now have 
shd /3’ 
3 _ 2 ^ 3 _ 1 - 13 cos 2cf> _ Jl 
^ sin^ /3 ’ 1 — yS sin^ ^' 
The expression for 2 ^ 0 , the perpendicular from the centre on to the tangent plane 
at 9, (f), is given in (49) of “ Harmonics,” namely. 
0/0 1\/ O l“f-/3 
ly 
/,;2 
0/0 1 \ / O 
2 Ar(v - 1 ) IV - 
/ O 0\ / 0 
^ 1-/9/ cos- yS COS- 7 1 
1 - y3 cos ^ “ Rin3 y 3 Ay ry * 
1-/3 
. (4). 
Also l)y (50) of “ Harmonics” the element of surface dcr of the ellipsoid is given by 
1 — /3 cos 2(f) 
^ 7-%, G, 3 _ 1 M 4- - 
(W dcj) 
1-/3 
- /X- 
P0Z3L 7.3 / 3 1 \" f 3 ^^4" _ 
yj \ )- l-fs) / I - /3 cos 2,f) \i (I + B _ , 
1 - yS ; \ 1 - /3 
7*' 
Passing to the new notation this may be written 
da m / k Y 1 - sild 0 - k'-^ cos 2 (j) m / k Y Ayry- /1 i ' 
d.e dcf) ~ Ttt/j V ^-0 / AF “ iirp \ /VQ / ■ AF siid 7 \Fy Ay I ’ 
VOL. CC.—A. 2 L 
