Solid Bodies through a Liquid. 339 



of motion, which for this slight modification need not be written 

 out again. W might be directly defined as the whole quantity 

 of work required to remove the movable solids, each to an infi- 

 nite distance from any other solid having a perforation with cir- 

 culation through it; and, with this definition — W may be 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 

 communication 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 yfr, cf>, 6 the rectangular coordi- 

 nates [x, y, z) of the centre of the movable globe. Then, because 

 the cores, being infinitely fine, offer no obstruction to the mo- 

 tion of the liquid, making way for the globe moving through it, 

 we have 



Q = im(^ + </ 2 +i 2 , . . . . . (25) 



where m denotes the mass of the globe, together with half that 

 of its bulk of the fluid. Hence 



^3 = ^=0 ^=0 



dx ' dy ' dz ' 



and j. / dQ 



^°( = ^) = m ^ ^0=^ ^m'z. 



(26) 



*l**(z 



A further great simplification occurs, because in the present 

 case etdyfr + 0d<j> + . . . , or, as we now have it, ctdx -f- fidy -f yds 

 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 



d . d . d\ 1 

 dx +y dy +Z dz~jD 3 



where r denotes the radius of the globe, and 



j) = {( x -a)*+{y-b)*+{z-c) <2 }K 



* Which means that if the globe, after any motion whatever, great or 

 small, comes again to a position in which it has heen before, the integral 

 quantity of liquid which this motion has caused to cross any fixed area is 

 zero. 



Z2 



