677* 
of Edinburgh , Session 1871-72. 
where r denotes the radius of the globe, and D = {(x - of -+• (y - h) 2 
+ (z-c) 2 p. Hence if N denote the component velocity of the 
liquid at ( a , b , c) in any direction A, fx, v , we have 
. d N 
where 
F 
N = (4 + 4y + 4) P C) > (27) ’ 
(*• * *> “> l ’ c ’> = i ,,3 ( X Ja + 4b + "Jo): 
A1 
D ’ 
Let now (a, b, c) be any point of the barrier surface O (§ 2), and 
A, fx, v , the direction cosines of the normal. By (2) of § 2 we see 
that the part of x due to the motion of the globe is ffNdo-, or, 
by (26), 
(4 + 4y + 4i)fP <*> * *’ °> C > ^ 
Hence, putting 
(28). 
(29). 
ff¥ (a?, y, *, a, 5, c) dcr - U , 
we see by (8) of § 4, that 
_dU _dU _ dE 
a dx’ ^ dy ’V dz 
Hence, with the notation of § 7 (18) for x, y,... instead of 9,... 
{y, *} = 0, {z, x] = 0, {x, y} - 0. 
By this and (25) the equations of motion (22), with (24), become 
simply 
d 2 x bW d 2 y bW d 2 z bW /om 
X + -r-, = Y + m^ 2 = Z + (30). 
dt 2 
dx ’ dt 2 
These equations express that the globe moves as a material particle 
of mass m, with the forces (X, Y, Z) expressly applied to it, would 
move in a “ field of force,” having W for potential. 
12. The value of W is of course easily found by aid of spherical 
harmonics, from the velocity potential, P, of the polycyclic motion 
which would exist were the globe removed, and which we must sup- 
pose known : and in working it out (small print below) it is readily 
seen that if, for the hypothetical undisturbed motion, q denote the 
fluid velocity at the point really occupied by the centre of the rigid 
globe, we have 
W = | fxq 2 -f w 
(31), 
