1884.] of the electromagnetic field. 151 



to the line whose direction cosines are proportional to d 2 L, b 2 M 

 and c 2 N. 



From this it follows easily that if we construct the ellipsoid 

 d 2 x 2 -\- b 2 y* + c 2 z 2 = 1, and take any radius vector length r as the 

 direction of L, M, N then this direction and the direction of the 

 magnetic force are axes of the section of the ellipsoid which is 

 formed by the plane passing through them; while any line in the 

 plane through L, M, N at right angles to this plane section may be 

 the wave normal. Thus so far as the equations hitherto considered 

 are concerned, there may be for any given possible direction of 

 electric displacement L, M, N an infinite number of possible wave 

 normals all lying in the plane 



|(6 2 -c 2 )+f (c 2 -a 2 )+^(a 2 -6 2 ) = 0, 



or for any given wave normal an infinite number of possible direc- 

 tions of electric displacement lying on the cone 



i (6 2 - c 2 ) + - (c 2 - a 2 ) + - (a 2 - b 2 ) = 0. 



x y z 



The value of V is given in equations (35), a simpler form is 

 obtained as follows. We have seen above, that \if=BL sin B, etc., 



then a = 4nrVB(mN-nM) sin 8 (44), 



etc. 

 <£> = — 2\B cos e cos S. 



Comparing this value of a with that given in (42), we have 

 Tr2 b 2 Mn-c 2 Nm 

 V = Mn-Nm =et ° (4o) ' 



Now, if p is the length of the perpendicular on the tangent 

 plane, at the extremity of a radius vector of length r in direction 

 L, M, N, then the direction cosines of p are prd 2 L, etc., and 

 q 2 = 1/pr. 



The direction of the magnetic force is, we have seen, at right 

 angles to this perpendicular, to the line I, m, n, and to the line 

 L, M, j\ r ; e is the angle between I, m, n and L, M, N, % that 

 between I, m, n and the perpendicular. Thus we have, expressing 

 the direction cosines of a, (3, y in the three possible ways, 



Mn -Nm _ pr (b 2 Mn - c 2 Nm) _ prMN (b 2 - c 2 ) 



sin e sin £ sin (e — £) '"{''' 



Also p = r cos (e - £) (47). 



