£90 The Rev. Professor O'Brien ow the Latios 



various metals, so that the light may be plane polarized after 

 two reflexions, or three reflexions, or four reflexions, &c. 

 In other words, he has determined a set of angles of incidence 

 which respectively make 2 « = 180°, 3 < = 180°, 4< <= 180°, &c. 

 He has also determined the inclination of the plane of pola- 

 rization of the twice-reflected ray to the plane of incidence, but 

 I do not know whether he has done the same in the case of 

 the rays reflected three times, four times, &c. ; there would be 

 of course no difficulty in observing the respective inclinations 

 of the planes of polarization of the rays which have been plane 

 polarized by three or more reflexions. Supposing this done, 

 it is clear that we obtain a series of values of (^ which re- 

 spectively make » = 90°, j = 60, j = 4'5°; and at the same time 

 we determine the corresponding values of Q, for (if we sup- 

 pose for simplicity that tan~^ — = 45°, or h — a) the corre- 

 sponding angles of inclination of the planes of polarization will 

 be tan-l (Q^), tan-^ (Q^), tan"! (Q^), &c. Sir D. Brewster 

 has found that there is only one value of ^ that makes < = 90°, 

 but that there are two values which make j = 60°, two which 

 make < = 45°, &c. Hence we can determine by experiment 

 several sets of corresponding values of ($, < and Q. 



6. Suppose then that we know the values of Q, » and ^ ; 

 substituting these values in the expression for U^, we may put 

 U^ in the form 



A' B 



where a and /3 are known numbers, and ^^ = "a~ W' ^^^"^^ 



the last equation in art. 3 becomes 



c2(a + ^V=l)2 + 2;>c(«i/3:i-/3) = |(l-^V-l), (1.) 



p being also a known number. 



Let «^, /3^, p^ be another set of known values of a, ^^p ; then 

 we have, similarly, 



c'^(«, + ^, V—\f + 2pp{a, v/^-/3/)' = |(l-^' ^^)- (2.) 



Equating the possible and impossible parts of the first mem- 

 bers of these two equations, we find 



c^ a /3 -(- /} a c = c^ a^ /3^ +JPi «/ <^) 

 which give 



Thus c becomes known, and consequently, by equating the 



