On the Intensity of the Light reflected from a Pile of Plates, 483 



by 2 cos a, we have 



(^m) 2 =l — 2cosa.<£(» + (0m) 2 (8) 



Squaring (7), and eliminating the function \p by means of (8), we 

 find 



\l—<j>(m)<p(n)\ 2 \l— 2 cos cc.<f)(m + n) + [(l)(m+n)'] 2 \ 

 = \l — 2 cos a. (p(m) + ((f>mY\ {l— 2cosa . <$>(n) + (<pnf\ . (9) 



From the nature of the problem, m and n are positive integers, and 

 it is only in that case that the functions <{>, \p, as hitherto defined, 

 have any meaning. We may, however, contemplate functions <p, ^ 

 of a continuously changing variable, which are defined by the equa- 

 tions (6) and (7) ; and it is evident that if we can find such functions, 

 they will in the particular case of a positive integral value of the 

 variable be the functions which we are seeking. 



In order that equations (6), (7) may hold good for a value zero 

 of one of the variables, suppose n, we must have 0(O)=O, \p (0)= 1. 

 The former of these equations reduces (9) for n=0 to an identical 

 equation. Differentiating (9) with respect to w, and after differen- 

 tiation putting w=0, we find 



<j)'(0)<p(m) { 1 — 2 cos a . <j>(m) + (<pm)H + cos a . $ (m) — 0(m)<//(m) 

 =cos a . <j> f (0) 1 1 —2 cos a . <f>(m) + (<$>mf\ , 



or dividing out by <p(m) — cos oc, (for 0(m)=cosa would only lead to 

 ^>(m)=cosa=0, \jj(m)=zC m 3 ) 



<p'(m)=<t>'(Q){l—2cosc(.((>(m) + (<pmy\. . . . (10) 



Integrating this equation, determining the arbitrary constant by 

 the condition that <p(m) = when*m=0, and writing /3 for sin cc . ^'(0), 

 we have 



sin (a+m/3) 

 Substituting in (8) and reducing, we find 



a m y= sin2 a . ...... (12) 



l * ; sin 2 (a+m/3) V ' 



But (8) was derived, not from (7) directly, but from (7) squared; 

 and on extracting the square root of both sides of (12), we must 

 choose that sign which shall satisfy (7), and therefore we must take 

 the sign +, as we see at once on putting m=n = 0. The equation 

 (12) on taking the proper root and (11) may be put under the form 



<f>( m ) = W m ) = l . .... (13) 



sin (mj3) sin a, sin (a-|- mfi) ' 



and to determine the arbitrary constants a, /3 we have, putting m—1, 

 and <p(m)=r, $(m)=t, 



J^=J-= . ,* (H) 



sin (3 sin a. sin (a + (3) 



