G9G MR. G. F. FTTZCtERALD ON THE ELECTROMAGNETIC THEORY 
or the two equations 
S i<f> Q V v 9i=S ifa V v 9i 
S j<f> 0 F v 9 1 =Sy c/j (J Fv9i 
which are the same as can be got at once by putting / = 1 m—n —0 in the Cartesian 
superficial equations, and taking and Sp as arbitrary and independent, when we get 
d[J_dU 
d/° df v dg° dg l 
If now we assume that the axis of x is the line of intersection of the plane of 
incidence and the surface, we may evidently assume 9t 0 to be the resultant of an 
incident and reflected ray, and 9t : to be the resultant of the two refracted rays, so 
that we may write 
%=joVo -f/oVo =h r n +/ 1V1 
when jo,j'o,ji,j\ are respectively unitvectors parallel to the direction of magnetic dis¬ 
placement in the incident, reflected and each refracted ray. Similarly calling 1c 0 , Jc' 0) 
h lt k\ the unitvectors normal to these wave-planes, and z 0 , z' 0 , z v z\ the variable distance 
along these, evidently we may assume v 0 =£ 0 yy, V ' () =¥ 0 ~, &c., and substituting 
dz 0 dz 0 
these we at once obtain 
f».S£^ 0 +^.SiV v o=J-S»'M+|!;-SiM 
'^s^ 0 ; 0 +|i.s 7 v 0 =A!.si^ 1 +^.s^/ 1 
dz 
d h 
dz\ 
In these I shall now assume <£ 0 = 1 as it is convenient to suppose this medium to be 
isotropic and the velocity of propagation in it unity, and i {) and i' 0 are the directions 
in the wave-plane perpendicular to the magnetic displacement; so that if a 0 , /3 0 , y Q , 
a ! 0 , /3' 0 , y q be the direction angles of these lines referred to the superficial axes 
Si(f) Q i 0 =cos a 0 , S^ v 0 =cos a' 0 , $j<fi 0 i 0 =cos /3 0 , Sj(f> 0 i' 0 =cos /3' 0 , 
and if s -1 be that axis of the section of Sp</>p=l by one of the refracted wave-planes 
which is perpendicular to the direction of magnetic displacement, it is evidently the 
velocity of propagation of this wave in this medium, and we may write 
^=U5=s bTs.*. ^' 1 =(/)S bTs=rT5 
when v~ l is the perpendicular on the corresponding tangent plane, and consequently 
Si<f)ii=TvTs. cos a, 
