594 Professor CHALLIS, ON A THEORY OF LUMINOUS VIBRATIONS. 



— — = c . — — + c- - . -^ + - . —^ (b, 

 dt dz' \f dx- / dy'l ^ 



which equation resolves itself into the two following. (See Art. 3), 



d'f <r-f 



h . --J-+ I J+ i,ef = 0, (24), 



da;- dy 



^-c'^-J-.w-0 = o (25). 



The former of these equations is the one it was required to obtain. By reasoning like that bv which 

 equation (l8) was derived, the analogous integral of equation (24) is, 



/7 , /e 



f = A cos 2 S/ - X + A' cosZ V rV- 

 h I 



Hence it appears that a ray of common light cannot be transmitted in the medium so long 

 as h and I are different quantities. Hence also two rays of opposite polarizations cannot in 

 general be transmitted in the same direction with the same velocity, for in that case they would, if 

 they were equal, be equivalent to a ray of common light. But equation (25), integrated in the 



same manner as equation (11), gives for the velocity of propagation, c \/ \ + — , which, if — - 



be equal to the constant k, is the same for rays of opposite polarizations. In explanation of this 



apparent contradiction, it is to be said that if -^ = k, and consequently e = —^ the value of/ 



2ir /k 

 for a ray polarized in the plane of xz is cos — sj — x , which is not independent of A, and 



therefore not independent of the nature of the medium ; whereas experience shews that a polarized 

 ray remains the same under all circumstances, and is in no way affected by the medium through 

 which it passes. That the value of/ may be that which belongs to a polarized ray, we must 



have X\/h = \' the breadth of the wave; or, — = — . But the velocity of propagation corre- 



A C 



spending to \ is c'v 1 + k. Hence the time of vibration of a given particle, or the colour 

 of the light, remaining the same, the velocity of propagation must be altered in the ratio of 

 X' to X, and consequently becomes a \/l + k. This result was obtained by somewhat different 

 considerations in Art. 8. of my Paper on Double Befraction. 



J. CHALLIS. 



Cambridge Observatory, 

 March 2, 1848. 



