556 



which the polarization is a maximum. We have, from (25), (22), 

 and (24), 



dfa = (sin 2 cos a + sin 2 a cos 0) da) = 2(m 1 ) cos <r sin 2 da>, 

 and cos a is negative. Hence % decreases as w (and therefore 1} de- 

 creases, or as m increases. For n=l, z=n and ^=^-2; for 



wz= oo, cos <r=0, and therefore x 1 == 0, or the maximum polarization 

 tends indefinitely to hecome perfect as the number of plates is indefi- 

 nitely increased. 



For a given number of plates the angle of maximum polarization 

 may be readily found from (24) by the method of trial and error. 

 But for merely examining the progress of the functions, instead of 

 tabulating i for assumed values of m, it will serve equally well to 

 tabulate m for assumed values of i. The following Table gives for 

 assumed angles of incidence, decreasing by 5 from 90, the number 

 of plates required to make these angles the angles of maximum pola- 

 rization of the transmitted light, and the value of x x , which determines 

 the defect of polarization. 



f= 90 85 80 75 70 65 60 5640'(=ar) 

 m= 1 1-330 1-944 2-913 4'921 9'775 30-372 oo 



Xl ='433 -422 -390 -337 '265 -177 '075 



V. " On the Theory of the Polyhedra." By the Rev. T. P. 

 KIRKMAN, M.A., F.R.S. 



This is a revised version of a Paper having the same title, read on 

 the 30th of May, 1861, of which an abstract has been already given 

 at page 2 18. 



January 30, 1862. 



Major-General SABINE, R.A., President in the Chair. 

 The following communications were read : 



I. "On the Calculus of Symbols." Second Memoir. By 

 W. H. L. RUSSELL, Esq., A.B. Communicated by ARTHUR 

 CAYLEY, Esq. Received January 7, 1862. 



(Abstract.) 

 This memoir is the continuation of one on the calculus of symbols 



