1887.] Equations to the Motion of Perforated Solids. 125 
If therefore we put 
CoS E,/x,p ry [ Ldo,, B, = ,/*,p, Vite ¢,/#,p: 
a, ae r,/*,p = [ew ar m,2) do,, b, = b,/K,P; C, a v,/K,p, 
we obtain 
; dv 
Ys lens do, Se BY, i ee — 4p, — bq, ae, C,7, 
1 d& 
+ p dk, ecevecvvece (19) 
Now if JZ be expressed as a quadratic function of all the 
a 
momenta == = 
p dk 
dz 
But 2T = tu + IP = Du(X — X) + wwe a0o0.0U000 (20), 
by (5). Hence in order to obtain y, we must differentiate (20) 
with respect to x, on the hypothesis that the momenta X are 
constant, and that w is a function of «,; whence by (4) and (6), 
dT dd du 
2 dk. a Gi, Gin, © (2,04, ae BY, + YU, + O,p, + 6,9, a C,7,)p 
i 1 
NG d 
a ee (21). 
1 
From (5) we obtain 
0 = 2% (u,v) + ap—p [| do, 
eeececcecreseseeecoos ee eoeseoeee eee ees eseee 
eopeseeceesesoceceteeceseeeseeoesoe ee 1 eee OD 
where the summation extends to all the unsuffixed letters in- 
cluding v=u, Multiplying these equations by w,, v,... respec- 
tively and adding we obtain 
dep ray 
= cP a + (au, + By, +7,w, + a,p, + 6,9, + ¢,7,) p— ale dco,=0, 
whence by (19) and (21) ae = pv, 
whence V,= wh. 
Hence the flux through the aperture o, relative to the solid 
S, is the generalised velocity corresponding to the momentum «,p. 
6. As an example of the above formule, consider the motion 
of an anchor ring discussed ante p. 47. 
