86 JAMIN ON METALLIC REFLEXION. 



y = J sin « COS (2 7r=^ + S j, vibration perpendicularto plane 



of incidence. To abbreviate, we shall put 



I cos a 



- — : — = cot « ; 



J sin a 



and there results, neglecting a constant factor, 



t 



00 = cos a cos 2 TT ^, 



y = sin a cos 



(24+8) 



(10.) 



The elimination of the time between these two equations will 



give the equation to the trajectory, which is an ellipse whose 



equation is 



w^ X- 2 cos 8 . „ 5, 



-7^ 1- — 2 -. xy = sin^ 6. 



sin'' a. cos^ a sm « cos a 



To obtain at the same time the direction and length of the axes 

 of the ellipse, we have only to replace the co-ordinate axes by 

 another system, making an angle («) with that to which the 

 equation is referred, and to take the condition that the coefficient 

 of (a? y) may disappear ; we shall then have the equation of the 

 ellipse 



(sin^ a sin^ oa + cos^ « cos^ w + 2 sin a cos a sin co cos w cos 8) y^ 

 + (cos^ a. sin^ CO + sin^ a cos^ co — 2 sin « cos a sin co cos 00 cos 8) x'^ 

 = &c., 

 and the equation of condition 



tan 2 CO = tan 2 « cos 5 (11.) 



This latter gives us the direction of the two axes at the same 

 time ; and replacing (co) by its value in the coefficients of y'^ and 

 a?% we shall obtain numbers proportional, the first to the axis of 

 (<r), and the second to the axis of (y). 



We shall put 



A^ = sin^asin^ CO -f-cos^otcos^ CO + - sin^asin^cocos8...axisofa?l 



.. 1 . . k'-^ 



B^ = sin^«cos^co-f-cos^asin^co— -sin2«sin2cocos8...axisofyj 



Let us now direct this elliptically polarized ray, or which 

 comes to the same thing, the two rectangular vibrations (10.) 

 upon a doubly refracting prism, making an angle (co) -with the 

 plane of incidence x : we shall have, calling (a/) the vibration 



