256 
A consists of double integrals, and gives the conditions at the 
limits. 
The differential equations of motion derived from (8) are 
(no external forces x, ¥, z acting) : 
aE dy & 
TS — Te = Q) Te Se (9) 
€ 
These equations will admit of the particular integral 
~ = cosa. f(w), n=cosP.fiw), C= cosy-flw), 
w = lx + my + nz — vt, 
provided it is possible to satisfy with real values of (a, (3, y, v) 
the equations of condition resulting from the substitution of 
these values in the equations of motion. 
These equations of condition lead to the following con- 
struction for the directions of the possible vibrations of mole- 
cules, and the corresponding velocities of wave-planes. 
Construct the six fixed ellipsoids, 
p= av?+ ny? + M2? + 2ayz + 2aaz + 2a;zy = 1, 
q= BY + L2 + Na? + 28,yz + 28.0724 2B,2y= 1, 
R= C2’ + Ma? + Ly? + 2yiyz4 2yrz 4 2ys2y = 1, 
Fo a2? + By + yz? + 2Lyz 4+2y3;02 + 2B.2y=1, 
G = agt* + Boy? + yor + 2ysyz + 2Maz + 2Zazy = 1, 
H = a30? + B3y’ + y32? + 2B.y2 + 2azz + 2Nay = 1, 
(10) 
and from their common centre draw the normal to the wave- 
plane, this will pierce the surfaces in six points ; let the corres- 
ponding radii vectores be p,p,, p,,, 7,71, 7,3 With these con- 
struct the ellipsoid 
a ag Mae ean 
Py Pu Pan 
2 2 2 
yz wz, 2ey 
= ig eS SS (11) 
2 
S “l “Mi 
The axes of this ellipsoid will be the three possible directions 
