20 Mr GREEN, ON THE REFLEXION 



By means of the last four equations, we shall readily get the values 

 of F"'x" f"x" m terms of f", and thus obtain the intensities of the 

 two reflected and two refracted waves, when the coefficients A and B 

 do not differ greatly in magnitude, and the angle which the incident 

 wave makes with the plane surface of junction is contained within 

 certain limits. But in the introductory remarks, it was shewn that 



A 



■= = a very great quantity, which may be regarded as infinite, and 



therefore g and g t may be regarded as infinite compared with 7 and y r 

 Hence, for all angles of incidence except such as are infinitely small, 

 the waves dependent on <p and <p t cease to be transmitted in the regular 

 way. We shall therefore, as before, restrain the generality of our 

 functions, by supposing the law of the motion to be similar to that 

 of a cycloidal pendulum, and as two of the waves cease to be transmitted 

 in the regular way, we must suppose in the upper medium 



\J, = a sin {ax + by + ct + e) + /3 sin ( — ax + by + ct + e), 

 and (20). 



(p = e ax (A sin \f/ + B cos \J/ ) ; 



and in the lower one 



v// = a sin (a x + by + ct), 



(21). 

 <t>, = e a,x {A, sin ^ + B t cos ^ ), 



where to abridge \// = by + ct. 



These substituted in the general equations (14) and (15), give 

 c« = 7 2 {a* + P) = 7; {a? + b>) =g*(- a' 2 + b 2 ) =g?{- «/ 2 + b 2 ), 

 or, since g and g, are both infinite, N 



b = a' = a;. 



It only remains to substitute the values (20) (21) in the equations 

 (17), which belong to the surface of junction, and thus we get 



bA sin \//o + bB cos ^ + ba cos (>// + e) + bfi cos (\|/ + e) 



m — bA t sin \f/ — bB t cos \// + ba / cos >|/- » 



