On Systems of Rays. 129 
mal velocity of propagation, so that its components are —w’da, —wy, —wdz: that 
is, we must have the equations 
: (w* —a’) da = Z (w —b*) dy = : (w* —c”) 82, (E"*) 
v 
in which o, 7, v, are, as before, the components of normal slowness, so that the equa- 
tion of the wave-element containing the transversal vibration is 
cou + Toy +voz=0. (A"”) 
These equations (A) (F'") suffice in general to determine, on FRESNEL’s princi- 
ples, the velocities of propagation and the planes of polarisation for any given wave- 
element in any known crystallised medium. 
Thus, eliminating the components of displacement 8x, dy, 8z, between the equations 
(4) (F'*), we find the following law of the normal velocity w, considered as depend- 
ing on the normal direction, that is, on the ratios of «, 7, v, 
= + tote =0. (G") 
wae to 
To deduce hence the direction and velocity of a ray, for any given normal direction 
and normal velocity, compatible with the foregoing law, that is, for any given values 
of the components of normal slowness o, 7, v, compatible with the relation (G"), we 
are to make, by (JZ), 
Pie (Q ar 1 DE 18 
ee o+72+u% ” CH ) 
and we then find, by (7), or by (D"), the following expressions for the components 
of the velocity of the ray, 
if we put for abridgment 
Ps eee 
e 2 2 2 
(Gem)? (gem) tae) 
And to deduce the law of the velocity + of the ray, considered as depending on its own 
2 
(K") 
direction, that is, on the cosines a 3 y of its inclinations to the semiaxes a 6 c of elas- 
ticity, we are to eliminate (according to the general method of the second number) 
VOL. XVII. 2L 
