671 
of Edinburgh, Session 1871-72. 
where Q, (IJ 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 +3) (^ + J + 1) ? which is equal 
A 
to the whole number of the coefficients 
i (* + i) , j (j + i) 
+ 
of the 
2 ' 2 
two quadratic functions, together with the i j available quantities 
* 
Ctj ^, . • • (X , ^ ... 
4. The meaning of the quantities a, a',... thus introduced 
is evident when we remember that 
dT : dT 
d£ 
drj 
dT . dT 
~ P ’"' <Ik ~3Tk'~* r 
( 6 ). 
For; differentiating (5), and using these, we find 
; dQ . dQ 
f = Tit > 9 = 
d( 
dr,’"' ' 
( 7 ), 
and using these latter, 
X = -gr;- ai A ~ Pfi ~ &c o X = zt - dif/-/3'<p - &c.,... (8). 
dn' 
Equations (8) show that - f3(p, - a'if/, &c., are the contribu¬ 
tions to the flux across fi, 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, *',-•• are applied to the 
liquid at the barrier surfaces, we must apply to them impulses ex¬ 
pressed by the equations 
£ = clk + aV + &c., r] = /3k + (3'k + &c.,... . (9). 
5. To form the equations of motion, we have, in the first place, 
bT 
«*x’ 
= 0... 
( 10 ), 
I 
and therefore, by (1), 
