677 * 
of Edinburgh, Session 1871 - 72 . 
where r denotes the radius of the globe, and D = {(cc - a ) 2 + (y - bf 
+ ( z-c ) 2 }L Hence if N denote the component velocity of the 
liquid at («, b , c) in any direction X, /x, v, we have 
where 
N = 
> y> z 7 a, h c), 
(27), 
F (x, y, z, a, b, c,) 
d d 
da + ^db 
Let now (a, b, c) be any point of the barrier surface £2 (§ 2), and 
X, /x, v , the direction cosines of the normal. By (2) of § 2 we see 
that the part of \ due to the motion of the globe is ffNda, or, 
by (26), 
( x dx + V 
Hence, putting 
ffF (x, y, z, a, b , c) da = U, 
we see by (8) of § 4, that 
d . d 
dy + v dz 
(*> y> z > a> h ’ c ) da c 28 )• 
rfU _ dll _ dV 
a ~ dx’ l3 ~ dy’ y ~ dz 
Hence, with the notation of § 7 (18) for x, y : ... instead 
Of ij [/ y • d o 
{y, z] = {*, x ] = { x : y} - o. 
By this and (25) the equations of motion (22), with (24), become 
simply 
m 
d 2 x 
dt 2 
X + 
bW 
dx ’ 
m 
d 2 u 
dt 2 
= Y + 
bxv d 2 z 
dy dt 2 
= Z + 
bxv 
(30). 
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 XV for potential. 
12. The value of XV 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 
XV = \yq 2 + w 
( 31 ), 
