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, Y, z acting) : 



d-'l d:'n d?Z ,^. 



^^ = ^- ^^ = «" ^^ = "^- (^> 



These equations will admit of the particular integral 

 ^ = cos a .J{w), rj = cos j3 •/(<«>)» ^ = cos 7 ./{oi), 

 01 ■= Ix -\- my + nz — vt, 



provided it is possible to satisfy with real values of (a, j3, 7, 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 sixjlxed ellipsoids, 



p zz Aa;2 + Ni/2 + Mz^ -\- 2aiyz + 2a2XZ + 2a3xy = 1, 

 Q= By' + hz'- -f Na;3 + 2^,yz + 2^^xz-\- 2^,,xy- 1, 

 R = cz^ -}- Ma;' 4- L2/2 + 27,2/2 -\- 2y^xz + ^y^xy — \, 

 F — aix^ + |3i«/' + 7i2' + 2Ly2 + 273a;2: -)- 2^.^y = 1 , 

 G = a^x^ + (i.2y^ + 723' + 2y3yz + 2mxz + 2aixy — 1, 

 H = a-iX^ + /Ss?/' + 732' + 2^.2yz + 2aixz + 2vixy = I, 



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 he p^p„ p,^,, r,r^,r„/, with these con- 

 struct the ellipsoid 



«' , r , 2' 2yz 2xz 2xy_ 



7r+ — + ^+7^ + 7i-+-— - 1- (11) 



p / p // P /// ' I '11 'ill 



The axes of this ellipsoid will be the three possible directions 



