698 
MR. G. F. FITZGERALD ON THE ELECTROMAGNETIC THEORY 
and the equations become 
A„(/„+/' 0 )=(«A 1 +mH 1 +«G 1 )|J +(rA' 1 +m'H' 1 +«'G' 1 ) f) 
A 0 (fl'o+A) = LA 1 +m 1 H 1 +« 1 G 1 )^ +(/' 1 A' I +m',H' 1 +»' l G' 1 ) 
Now if we consider A x , H ]5 G i we see that they are proportional to the direction 
cosines of the perpendicular on the tangent plane to U where it is met by g=0, h= 0 
— i.e., it is the direction conjugate to the electric displacement corresponding to rj,. 
Now if we call A () .? the velocity of propagation of this wave and A 0 s' of the other, we 
see at once that a' 1 is an axis of the section of U=a constant by one wave-plane, 
and s'” 1 is an axis of the section by the other wave-plane, and if v~ l and v'~ l be the 
corresponding conjugate directions, and oq and a\, /3 1 and /3j the angles between these 
latter and the superficial x and y axes, and if a 0 , /3 () , a' 0 , /3' 0 be the corresponding’ angles 
between y 0 and y' 0 and these same superficial axes, we get exactly the same equations 
as before, and write 
dvo , d V o / (I Vi , / / / dv i 
— . cos a fl +77 . cos a n =vs . cos oq —- -\-vs . cos a , —— 
dz 0 u az 0 J 1 dz l dz l 
dy 0 
dz o 
dif o 
dVi 
cos . cos /3' 0 = vs . cos /3 1 — — \-v's . COS a 
dz. 
C]V\ 
dz\ 
In order to reduce these further we assume 
Vo = — T 0 . cosf . (t—z 0 ) tj' 0 =T'q COS rf (t—z Q ) 
A(, A/ Q 
Vi= TiCos^- (st—h) v\=T\ cos 2 ^{st—z\) 
and if our superficial axes are so placed that the axis of x is the intersection of the 
plane of incidence and the surface, and if i Q , i' Q , i\ be the angles the respective wave 
normals make with the normal to the surface, we evidently may write 
Zq—z cos i-\-x sin i z'o =z cos fi+z sin % 
z x —z cos i L -\-x sin i, z\=z cos i\-\-z sin i\ 
as it is easy to convince oneself that any terms involving y would be inadmissible, as 
they could not by hypothesis occur in z 0 , and the terms involving the time could not 
vanish out of our equations if they occurred in the others. Hence these wave normals 
are all in the same plane. When z= 0 our equations must evidently be true inde¬ 
pendently of the time and x, from which we see that no change of phase is possible in 
