484 Boyal Society : — 



We readily get from equations (13), 



i ;/ \^s \— sinasin{a + (wi + w)/3} 

 YK yvv J sm(a + »w/3)sin(a + w/3) 



,/ . \ ,/ N sin a sin??/3 

 ty{in + n) — 0(w?.)= — !- ; 



sin { a + (™ + n )P } s in ( a + m fi) 



whence the equations (6), (7) are easily verified. This verification 



seems necessary in logical strictness, because we have no right to 



assume a priori that it is possible to satisfy (6) and (7) for general 



values of the variables ; and in deriving the equation ( 1 0), the equations 



(6) and (7) were only assumed to hold good for general values of 



m and infinitely small values of n. 



The equations (13), (14) give the following ^wa^-geometrical 

 construction for solving the problem: — Construct a triangle of which 

 the sides represent in magnitude the intensity of the incident, reflected, 

 and refracted light in the case of a single plate, and then, leaving 

 the first side and the angle opposite to the third unchanged, multiply 

 the angle opposite to the second by the number of plates ; the sides 

 of the new triangle will represent the corresponding intensities in the 

 case of the system of plates. I say gwc^'-geometrical, because the 

 construction cannot actually be effected, inasmuch as the first side 

 of our triangle is greater than the sum of the two others, and the 

 angles are imaginary. 



To adapt the formulae (13), (14) to numerical calculation, it will 

 be convenient to get rid of the imaginary quantities. Putting 



V {(l+r+t)(l+r-t)(l + t-r)(l-r-t)\=X . (15) 

 we have by the common formulae of trigonometry, 



cosa=— !_ ; s ina=ZJl i£ . 



tr 2r 



whence, putting 



i(l + ,.^^ +A) = ^ ..... (16) 



we have 



e ^- la =cosa+ v — 1 sina=a Tl . 



It is a matter of indifference which sign be taken : choosing the 

 under signs, we have 



2rsina= — V^Ta, <?V~i«— a . 

 We have also 



cos/3=i±^, S Jn/3=r s in«=-^E^ 3 



no fresh ambiguity of sign being introduced. Putting therefore 



I(1+*W + A) = 5, (17) 



we have 



