applied to the Action of Magnetism on Polarized Light. 93 

 provided 



and (/i 2 -^ 2 )B = ?K 2 «c 2 AJ • ( ° ) 



Multiplying the last two equations together, we find 



(n*-m*a*)(n*-m*b*) = m*n l c*, . . . (151) 

 an equation quadratic with respect to m 2 , the solution of which is 



. (152) 



a* + b* + ^/(a a -A 2 ) 2 + 4fiV 



These values of rri 2 being put in the equations (150) will each 

 give a ratio of A and B, 



A_ fl 2 -^q: ^/( fl g-y)a + 4wV 

 B 2nc 2 



which being substituted in equations (149), will satisfy the ori- 

 ginal equations (148). The most general undulation of such a 

 medium is therefore compounded of two elliptic undulations of 

 different eccentricities travelling with different velocities and ro- 

 tating in opposite directions. The results may be more easily 

 explained in the case in which a=b ; then 



m 2 = -2^— !> and A=+B. . . (153) 

 a -j- nc 



Let us suppose that the value of A is unity for both vibrations, 

 then we shall have 



/ • nz \ / nz \ - 



*= cosfwf- I + cos(^-- ) 



V va 2 —nc 2 / \ va 2 + nc 2 / 



(nz \ . ( , 7Z-2T \ 



ft/ ^ zr ) + sin I w£ I . 



The first terms of a: and y represent a circular vibration in the 

 negative direction, and the second term a circular vibration in 

 the positive direction, the positive having the greatest velocity of 

 propagation. Combining the terms, we may write 



x = 2 cos (nt — pz) cos qz,~\ 



Y . . . . (155) 



where 



y = 2 cos [nt—pz) sin qz, 



and 



2v / fl 2 -nc 2 2Va*+nc*' 

 n n 



(156) 



These are the equations of an undulation consisting of a plane 



