676 * Proceedings of the Royal Society 
out again. W might be directly defined as the whole quantity of 
work required to remove the movable solids, each to an infinite 
distance from any other solid having a perforation with circulation 
through it; and, with this definition, — W maybe put for in 
the equations of motion without exclusion of cases in which there 
is circulation through apertures in movable solids. 
11. I conclude with a very simple case, the subject of my com- 
munication to the Royal Society of last December, in which the 
result was given without proof. Let there be only one moving body, 
and it spherical; let the perforated solid or solids be reduced to an 
infinitely fine immovable rigid curve or group of curves (endless, of 
course, that is, either finite and closed, or infinite), and let there be 
no other fixed solid. The rigid curve or curves will be called the 
“core” or “cores,” as their part is simply that of core for the 
cyclic or polycyclic motion. In this case it is convenient to take 
for ij/, <p, 0 , the rectangular co-ordinates ( x , y, z ) of the centre of the 
movable globe. Then, because the cores, being infinitely fine, 
offer no obstruction to the motion of the liquid, making way for the 
globe moving through it, we have 
Q ~ lm(sc 2 + y 2 + z 2 ) . . (25), 
where m denotes the mass of the globe, together with half that of 
its bulk of the fluid. Hence 
/"\ 
c lx ? dy ‘ dz ’ 
and 
= mx, rj 0 = my, £ 0 = mi 
A farther great simplification occurs, because in the present case 
a dif/ + /3dp + ..., or, as we now have it, adx + fidy + ydz, is a 
complete differential* To prove this, let V be the velocity- 
potential at any point (a, b, c) due to the motion of the globe, 
irrespectively of any cyclic motion of the liquid. We have 
V = if 
.d 
. d 
. d 
+ y — i - 
dx dy dz 
)B' 
* Which means that if the globe, after any motion whatever, great or 
small, comes again to a position in which it has been before, the integral 
quantity of liquid which this motion has caused t© cross an}' fixed area is 
zero. 
