476 Prof. Challis on the Theory oj Double Refraction 



the present argument it is not necessary to be acquainted with 

 explicit expressions for these functions. 



It results from the preceding reasoning that K cos cf) is the 

 ratio of the retarding force of the atoms to the actual accelera- 

 tive force of the aether. But the actual accelerative force is the 

 force due to the actual variations of density, diminished by the 

 retarding force. Consequently, 



b 2 dp _ ___ a 2 dp ~ b 2 dp 



pdx ~~ pdx ^ " pdx ' 



whence ^(1 + Kcos <£)=« 2 . 



The coefficient K will evidently vary as the density of the me- 

 dium. Hence if $ be the density, and K = H8, and if p be put 

 for the ratio of a to b, we shall have 



^ 2 -l = HScos</). 

 This equation involves the explanation of the different refrangi- 



bilities of the rays of light. For since cot (j>= -—, and «=/zZ>, 



we have , 2ira 



cos 9 = 



^47rV + ? 2 Xy 



and . 1 TT ^ Zira , N 



u? — 1 = 1-18. ■ . — -— n - , .... (a) 



which equation determines the relation between /j, and X. By 

 differentiation it will be found that 



dX_ 4ttV / H 2 S 2 2H 2 ^y \ 



dp. " q 2 \p 3 ' V(/x 2 - 1 ) 2 + G* 2 - If J' 



Since H 2 S 2 = (//, 2 —l) 2 sec 2 <£, this equation proves that the varia- 

 tions of X and /x have different signs. Consequently as X in- 

 creases fi diminishes ; that is, the rays of greater breadth are the 

 less refrangible. The course of our reasoning, which has thus 

 led incidentally to an explanation of the phenomenon of disper- 

 sion, now brings us to the more immediate consideration of double 

 refraction. 



Resuming the equation (a), we have, for the relation between 

 the partial variations of q and X with respect to fx, 



dq __dX q 

 dfi d/jb ' X' 



But lz — q\\ — j^ ) = ?(! ^-)) k being a positive constant. 



Hence d.e q _ _ 2f a*_ / f^f\ dq 



d/i /Jb qfju z \ a 2 J djju 



a 2 — fjb 2 e 2 dX 



+ 



Xju, 2 d/ju 



