OF THE REFLECTION AND REFRACTION OF LIGHT. 
697 
when oq is the angle between v and the axis of x, and evidently Ts represents the 
velocity of this wave, while Vvj is the direction of the ray. Hence our equations may 
be written 
dvo 
J C Wo 
cos a^-^+cos a.'^-jy=Tv.Ts. cos oq 
dz n 
ds' n 
dr)i 
dz x 
+TfT/. cos a\. 
dv\ 
d*\ 
cos /3 0 .~-\-cos p 0 ^?=Tv.Ts. cos /3^+Tv'Ts. cos /3'^ 
' dd n 
■dz l 
dz' 1 
These may be readily deduced from the Cartesian equations as follows : U 0 being 
isotropic it can be written = A 0 ( f 0 2 -f- g 0 2 + h Q 2 ), and is evidently unaltered by trans¬ 
formation, and if l : m : n and l x : m l : n x be the direction cosines of x and y to any 
arbitrary axis, we get 
Ao f^tf+nlf+n 
dh 
. 7 dV 1 . dJJ 1 , dU x 
and now, as these are linear, we may suppose the superficial disturbance in one medium 
to be due to an incident and reflected wave, and in the other to two refracted rays, so 
that we can write these as 
A o(/o+/ »)-(/ d/ + m d g + n 7j;) + [!' d/ + m d g +» A 
a / . , v /, dU x , dV 1 , dU\ , ( v dV\ , dU\ . , dU\ 
K(9o+9o ) = (^+ w h^+ni^j + (Z i ^+ W ii- 5 -+n 1 — 
df i 
dg i 
and as U x and U , 1 are supposed to be referred to arbitrary axes, we may suppose 
to be referred to such that p 1 is the only component— i.e ., to such that the direction of 
the magnetic force is the axis of y, that of z being normal to the wave-plane, and 
similarly y\ being the only component in ~U\ while the 2 axis in this case is normal to 
its wave-plane, and of course this will be a function of z x only and rj\ of z\ only, 
these being the corresponding ordinates. We thus obtain 
so that 
„ /> dr] 1 dr)' 
^=-1^ 4, a=-*7 
9i=9' i=/q=/fi = 0 
1 
1 
dV 
dW 
fHT 1 
dU' 
dU'j 
^ =Hl/l V =H ' l/ ' 1 ^ =Gl/l v 7= g,i/ > 
