in the Undulatoiy Theory. 169 



On this hypothesis the axis x passing through the first 

 molecule m in any direction, the sums of the corresponding di- 

 stances of all the other molecules on each side of it, whether 

 in the plane of y or z, will be equal for all positions of x in 

 the medium. 



It is easily seen that the respective sums of products 



(^ r sin 2 ^ \1/ ;• A j/^ sin 2 ^ r}/ r A x;- sin 2 ^ 



■i^ r Ay A s sin 2 ^ 



with opposite signs will be equal. Thus we shall always have 



1, [uq sin 2 0] = O'] 



^ lapsing 6] = y. (50.) 



S [a p' sin 2 6] = J 



Whenever these terms are evanescent, it is easy to show 

 that we always have also 



2 [aq2s\n^ 6] = 0; (51.) 



and similarly for the like terms involving /3, by a simple trans- 

 formation of coordinates, as explained by Sir J. Lubbock in 

 his valuable paper*. That paper indeed relates to the more 

 general views of the subject, to which I shall refer in the se- 

 quel ; but the particular process in question is independent of 

 these views. 



Thus the hypothesis of symmetrical distribution gives 



2 [y A r,] = S [? A ^] = (52.) 



and the original equations are I'educed to 



^ = t[pA,-] (53.) 



^ = S[/A?]; (54.) 



or restoring the original values, 



^ =S[(4>(r) + 4/(r)A7/=)A,] (55.) 



ili =2[(<f(r)+^^(r)A.=)A?]. (56.) 



That is, the equations are reduced to precisely the same 

 form by the hypothesis of uniform distribution, as they are on 

 the hypothesis of unsymmetrical distribution, by the condi- 

 tions of plane polarized and unpolarized light. 



Thus the formulas for plcnie polay^izcd and utipolarizcd light 

 are o?;/j/ solutions of the original equations when in the same 

 form, to which they are reduced by syinmctrical dhtrihution. 



* L. & E. Phil. Mag. and Journnl of Science, vol. xv. November, 1839. 



