14 Mr GREEN, ON THE REFLEXION 



and therefore 



1- ^ 



F[ a _ a - a, _ cot 9 - cot 9, _ sin {9- 9) # 



J" a t ~ a + a t ~ cot 9 + cot 9 t ~ sin (9 / + 9) ' 



a 



9 and 9 t being the angles of incidence and refraction. 



This ratio between the intensity of the incident and reflected waves, 

 is exactly the same as that for light polarized in the plane of incidence, 

 (vide Airy's Tracts, p. 356,) and which Fresnel supposes to be propagated 

 by vibrations perpendicular to the plane of incidence, agreeably to what 

 has been assumed in the foregoing process. 



We will now limit the generality of the functions f, F and f t , by 

 supposing the law of the motion to be similar to that of a cycloidal 

 pendulum; and if we farther suppose the angle of incidence to be in- 

 creased until the refracted wave ceases to be transmitted in the regular 

 way, as in our former paper on Sound, the proper integral of the 

 equation 



d i w l , j d 2 w t d 2 w} 



IF ~ 7/ 1 das' + If) ' 

 will be 



w, = e- a '' x Bsm^, (10); 

 where ^ = by + ct, and a' is determined by 



7 /(6 2 -«; 2 ) = ^ = y(* 2 +« 2 ), (ii). 



But one of the conditions (9) will introduce sines and the other 

 cosines, in such a way that it will be impossible to satisfy them unless 

 we introduce both sines and cosines into the value of w, or, which 

 amounts to the same, unless we make 



w = a sin {ax + by + ct + e) + /3 sin ( - ax + by + ct + e), (12), 



in the first medium, instead of 



w — a sin (ax + by + ct) + (5 sin ( — ax + by + ct), 

 which would have been done had the refracted wave been transmitted 

 in the usual way, and consequently no exponential been introduced into 



