671 
of Edinburgh, Session 1871-72. 
where Q, OJ denote two quadratic functions. This we may clearly 
do, because, if i be the number of the variables >7, — , and j the 
number of k, k'...; the whole number of coefficients in the single 
quadratic function expressing r is ^ which is equal 
A 
to the whole number of the coefficients + ^ 4- of the 
2 2 
two quadratic functions, together with the i j available quantities 
a, a , /5 , . . ... 
4. The meaning of the quantities a, (3,... a',... thus introduced 
is evident when we remember that 
dT . dT dT . dT 
d£ d v ~‘ P ’"' d K 
For ; differentiating (5), and using these, we find 
= 
dQ 
w 
dQ 
<V"‘ ' 
( 6 ). 
CO) 
and using these latter, 
X = .,#= -/¥?-& C ^,.. 
( 8 ). 
Equations (8) show that - a \p, - ft <p, - a'ij/, &c., are the contribu- 
tions to the flux across O, O', &c., given by the separate velocity- 
components of the solids. And (7) show that to prevent the solids 
from being set in motion when impulses k, k',-*- are applied to the 
liquid at the barrier surfaces, we must apply to them impulses ex- 
pressed by the equations 
£ — a k + aV + & C ., 7 } ~ @k + P'k + & C .,... . (9). 
5. To form the equations of motion, we have, in the first place, 
^-0 ^ -0 
d X ~ ’ dx 
( 10 ), 
and therefore, by (1), 
dK 
dt 
A K, 
dt c' T _, 
W = K ’ 
(ii); 
