16 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT 



SECTION III. 



Application of the Equations of Connection just obtained to the case of ordinary 



Reflexion and Refraction. 



(14.) We shall first consider the case of cylindrical or plane-waves perpendicular to the plane 

 of xx (which will therefore be the plane of incidence), the vibrations taking place at right angles 

 to that plane. 



In this case a = 0, 7 = 0, a'= 0, y'= 0, and /3 is independent of y : therefore the six equations 

 of connection, (Z)) and (/), Section 11, reduce to two, viz. 



We shall suppose that the whole motion consists of three sets of waves (for we shall shew 

 presently that it cannot in general consist of only two), one set in the upper medium, and two 

 in the lower. Let /3 + /3, be the whole displacement at any point of the lower medium, the part /3 

 arising from one of the sets of waves, and the part /3, from the other ; then we must write /3 + /3, 

 instead of /3 in the two equations of connection, which therefore become 



/3 + A=/3' (1). ii^-^l^)-'^ (^)- 



Now, using the notation in Article (2), we have 



^=F ^=r ^=F' 

 dt ' de '' dt 



Hence, and by the equations {A), Article (1), -j^ and (2) immediately give us 



Again, since by the equations {J), Article (I), we have 



dV^__pdV ^Si__Pj^ dF' p dV 



da> V dt ' dw v^ dt ' dx v dt 



and by (3), V+ V- r=0; 

 and smce -, — , — , are either constants (in the case of plane-waves), or vary very slowly 



compared with V, F , F', on account of the extreme smallness of the length of a wave of light ; 

 we have by Article (3), (see Note), 



^ = ^ = ?.' 



« v^ v' ' 



Now v = V, therefore ' 



P,= P (5), P = np' (6) jwhere M=^[' 



Hence, observing that q, q^, q, are each zero, we have 



