158 The Rev. S. Hauauton on the Equilibrium and 
from which the following value for v, may be deduced, by substituting for (p’) its 
value taken from (6), 
2. = a(Z) + (St) +e (sy + Lu? + mv? + nw? (11) 
+2(1 A Ag w) + 2(avw + pow + y,vv) 
+f(olbanlgent)(odhtadtindl)+n( dante 
and if we make dw = p'sin@dpdédq, we shall have the following values for the 
coefficients of the function v,: 
A = 2S r,.cos‘adu, B= 2\Sr,.cos'Bda, c = 2§F,.cos*y dw, 
L = 2§§r,.cos’Bcos*ydw, M= 2SSr,.cos’acos*ydw, N = 2(\\F,.cos*acos’Bdu, 
a, = 2\{SF,.cos*acosBcosydw, B,—= 2SSF,.cos*Bcosyda, y, = 2\\\F,.cos*y cosBdu, 
a, = 2§\SF,.cos*acosydw, B,=2\\Sr,.cos*Bcosacosy dw, 7, = 25S F,.cos*y cosadw, 
a, = 2§\\r,.cos*acosBdu, B,= 2SSr,.cos*Bcosadw,  y, = 2\\\F,.cos*ycosacosBde. 
It appears, therefore, that the form of the function y for solids (in which we have 
v=yV,), 1s that of a homogeneous function of the second order of the six quan- 
dé dy dg 
’ da’ dy’ dz 
tain twenty-one constants, but the function v, which has been just determined, 
tities, » U,V, W. The most general form of such a function would con- 
contains only fifteen constants, although it has the full number of terms. The 
general equation of equilibrium and motion of solid bodies is therefore 
§Sp (xbE ++ ¥en + 2é¢)dm = SS bv,dadydz, (12) 
in which the expression (11) for v, must be substituted, and the equation treated 
in a manner analogous to what has been already done for equation (5), this will 
produce the general equations of equilibrium and motion, and also the conditions 
at the limits expressed by means of double integrals. 
It is important to observe that the function v,, which has been just found for 
solids, is quite different from the function v, used by Professor M‘Cullagh in his 
Mechanical Theory of Light; in that theory the function v, which defines the 
medium whose vibrations produce the sensation of light, is a homogeneous func- 
tion of the second order of the three quantities, 
