1887.] Equations to the Motion of Perforated Solids. 119 
We know that the motion could be instantaneously produced 
from rest, by the application of suitable impulses to each of the 
solids and barriers. Let X,,, Y,, Z,; L,, Mn, Nim be the force and 
m ? m? 
couple components of the impulse along and about axes fixed 
in S,,, which must be applied to S,,. 
Let &,, n> Sn3 Nm» Mm? Ym &m> 1m--- be the components of the 
impulses which se be applied to each of the barriers of S,; also 
let €,= 28, &c.; =X,,+€, &, and let ¥,, 9,,, Z@ k Ag 
fH, P,,, be the Lae components corresponding to w,,, ¥,, « 
of the momentum of the cyclic motion, when all the solids are 
at rest. 
Let M,, be the mass of S,,, @ the kinetic energy of the liquid, 
T that of the whole motion. It will be shewn that 7 is the 
sum of two homogeneous quadratic functions of the velocities and 
circulations respectively. Let these be denoted by J and & 
respectively, and let (w,,%m), 2(u,,¥,) denote the coefficients of 
mm 
De, U0. &e. 
Me, m ~m 
At the surface of S,,, d®,,/dn is equal to the normal velocity of 
S,,. and is zero at the surfaces of all the other solids; hence 
o= ae z®,,; 
and therefore 
20 =—p |[ (T oe aS, ae : dS, SY. g ) 
val («,do, +«/do/ +... Kao, + a 
Hence 
(un) =- p[[ 4% as,, 
uu) =—p | [os a6, aS,— || “P* a8,=—2p||o, a 
| 
2 tne) =—p [foe AS, + ole dos= 9. | 
| 
| 
| 
| 
| 
J 
) 
2 (u,x.) =—p |e w, a dS, + alee do,=0, 
da 
(«,«,) =p [Frde,, 
‘ dw,’ doen , dw,’ 
2 (Ke; =p |[ Gedo, + pf | Pde, = 2p|] = do 
VOL, VI. PT. III. 9 
