130 Professor Hamitton’s Third Supplement 
the ratios of « + v between the three expressions (Z'*), and so to deduce the relation 
between the three components of velocity ~ 5 = : ; now the equations (J) give 
evidently, by (K"), 
aca? be Cty? 
A2—a? a A2—B + AL—e? 
=05 (i) 
they give also, when we attend to (@"), 
Mee B 2 oa a5 se M's 
Ne) shee oleae’ «2 ED 
A therefore is the velocity of the ray, or the radius vector of the curved wnit-wave, 
propagated in all directions from the origin of co-ordinates during the unit of time ; 
and the equation of the wave in rectangular co-ordinates x y z, parallel to the axes of 
elasticity, is 
2 
acre b2y2 et? 
Pipe + eye t aye =o OY 
or, when freed from fractions, 
(a? +4? +27) (aa? + By? +072") + 0B? 
HCV+ )P +0 (C+a)Yt+eaes+h)2. (O*) 
This method of determining the equation of Fresnet’s ave, will perhaps be thought 
simpler than that which was employed by the illustrious discoverer, and than others 
which have since been proposed. 
Reciprocally to determine by our general methods the normal direction and yelo- 
city, or the components of normal slowness o, 7, v, for any proposed direction and 
velocity of a ray compatible with this form of the wave, that is, for any values of 
a B y X compatible with the relation (L"), we are to substitute for the ray-velocity A 
in that relation its value (J/"*), and we find, by (#*), 
su_a 1—a% 
6a 0° =a? 
_ ov Lins, We bay 18 
ae aie Sok. 
Bay Oe he 
‘. sy v0” Hc? 
if we put for abridgment 
Ga) + Ge) + ES) 
