372 PROFESSOR CHALLIS, ON A THEORY OF THE POLARIZATION OF LIGHT 



Let us now consider by itself the polarized ray in which the condensation is tj, . Since 



rf(T2 ds dcr, d<7y ds rfcTi „ , . . . . 



4 = 0-, + (Ta, we have --— = — — — — , and - — = — — . Substituting these values in (16). 



we obtain, 



da; dx da; dp dy dy 



rf<r,-' ds dff, d<T\ ds da^ 

 d.c' dx' dx dy' dy' dy 



Also (ji must satisfy equation (l5). Consequently 



cPo-i rf'Vi ,, , , 



-r^ + -T-v + «'A;a, = (18). 



dor dy 



The equations (17) and (18) determine the function that cr, is of .r and y. For by eliminating 



■ from (18) by means of (17)) an equation results, which, as it contains only partial differential 



d.v' 



coefficients of a, with respect to y, determines the form in which y enters into this function. The 

 form in which ,r enters is similarly determined. The function expressing the value of u.^ is deter- 

 mined by equations exactly the same as (17) and (18), having only cr-, in the place of (t^. In fact, 



1 , p '^'''i" ,dij\ „ , ds d(ji da, , 



puttme equation (17) under the torm — A — — + i» = 0, we have A = —— = — — + — — , and 



^ dx dx dx dx dx 



„ rfcTi ,ds da:\ da-f da. da, da, r^, r < ■ ■ ,. 



B = I = — = — . . 1 he two roots oi that equation are therefore 



dy ^ dy dy j dy dy dx dx 



— ' and — ~ , and hence the process indicated above which determines cti determines 0-2 also. 

 d.v dx 



Since the original ray is supposed to be one of common light, « is a function of r, and 



d s , X d s . y „ , . . , , ■ . , V 1 



— = / (r) - , - — = f{r) . - . Substituting these values in equation (17), we have, 

 dx r dy r 



dx' dy' •' ^ ' \dx r dy rl ^ ^' 



If now the direction of the axes of co-ordinates be changed througli 90° by putting - x' for y and 

 y' for >r, neither this equation nor equation (18) will be altered in any respect, so that the same solu- 

 tion of the equations will result as before. Consequently by this transformation the function that 

 (7, is of X and y is converted into the function that a-^ is of the same variables, and rnce versa. It 

 follows that the original ray is divided into two equal polarized rays, such that if one be turned 

 about the axis of z through 90° it becomes identical with the other. Since also equations (18) and 

 (19) are not altered by changing the axes of co-ordinates through I80", that is, by altering the signs 

 of ,r and y, it appears that a, and u- are symmetrical about planes passing through the axis of J'. 

 These planes, from what has just been proved, must be at right angles to each other. Also it is 

 evident that a plane of symmetry of one ray must coincide with a plane of symmetry of the other. 

 Hence each ray will have two planes of symmetry at right angles to each other. 



The above results would be more properly derived from the functions of ,r and y expressing the 

 values of ai and (j.., if the integrations could be performed by which these functions would result 

 from ec|uations (IS) and (19). This it does not appear possible to do generally; but values of a, 

 and (7, applicable to small distances from the axis of z may be obtained as follows. We have seen 



that the solution of equation (13) for small values of r is/= cos }i V - »'. Hence, as s =/'^. we 



