1888.] in a Lnquid when the impulse reduces to a couple. 279 
Now let &, 7, € be the coordinates of the centre of the rota- 
tional ellipsoid referred to axes fixed in space, of which & is normal 
to the plane of the resultant impulsive couple, on which the 
rotational ellipsoid rolls, and 7, € are parallel to axes fixed in this 
plane, then we have 
E=Ddu+m 1,47 w | | 
N=rNU,+p VU, +, Ree sees (28), 
C=r'u, + py, +v"y, | 
where w,, v,, w, are given by (10). These equations become 
3 —— jade + DB? + oC"? +f (B+ C2) wv +g (0? + A”) or 
+h (A? + B) me ...(29), 
d ° r 2. 7 * or 2 / ey 2 , yy, 
= (0+ tb) => — | aA? (Nv + 1d") + DBw (wu! + tu”) + Ov (v’ + iv") 
| +f{Co(u' + te") + Bow (v' +")} + g {AP7A(v' +0”) 
+ O7v(M +in")} +h {Bee (MN +IN") +420 (us! +5p”)} 
[NP cael a pele: Lig ip ie Ss ee (30), 
and & (yn —2€) is obtained by writing —2 for 7 in this. 
11. To integrate the equation for € we have only to observe 
that each of the terms on the right is either an ordinary elliptic 
function, or else a doubly periodic function of the second kind of 
the same form as that on the left of (20). 
2. (Qu — ea) (OU — e) 
hus An— CPanel =e) 
x QU — eq ad 
(Ga ra en) (Ca a ey) du 
ee similarly for «* and v’ by cyclical interchanges of the letters 
a, 8, ¥. 
oe Ap is proportional to 
© (w—w,) © (u — weg) 
O*u 
(GPA HEE? bd hae (31), 
exp (Ma + NB) Us 
which is equal to 
60, Ow, d [6 (u—a, — ag) 
G (@, + @p) du 3 Gu kD (ta +78) | 
and uv, vd can be prepared for integration in the same way. 
