512 



The Bev. S. Haughton on some new Laws 



It is required to express in terms of the readings of the ana- 

 lyser and compensator, the position of the axis major of the 

 eUiptic polarization and 

 the ratio of its axes. This 

 may be accomplished by 

 the following considera- 

 tions. 



Let the elliptically-polar- 

 ized beam be conceived as 

 inscribed in a rectangle 

 whose sides are parallel 

 and perpendicular to the 

 plane of incidence 01. Let 

 Ox be the diagonal of the rectangle circumscribed, and Oy the 

 axis of the ellipse ; it is required, from the difference of phase of 

 the light in the planes 01 and OP, and knowing the direction of 

 the line Ox, to find the direction of Oy and the ratio of axes. 



It will assist our investigation and involve no hypothesis, to 

 imagine the ellipse as the path of a point revolving in the direc- 

 tion of the arrows, whose coordinates may be expressed as fol- 

 lows : — 



f=Asin (kt-{-e) 



7; = JBsin {kt'\-e'), 



where el—e is the difference of phase between the directions 01 

 and OP, and A and B are the lines 01 and OP. 



Eliminating t from the expressions for f and 77, we find the 

 equation of the ellipse, 



£ + g-3cos(e'-e)^= ,m\ef-e). 



(5) 



It is well known that if the equation of an ellipse be 



Da?« + 2E^2/ + F2/2=Q, (6) 



the position of its axis is found from the equation 



• tan2</,=:g— J,. 



Substituting for D, E, F their corresponding values in (5), we 

 obtain 



tan2</)= tan2acos(e'— e), .... (7) 



if} being the angle lOy, and a the angle lOa:. 



Returning again to equation (6), it is not difiicult to prove 

 that if a and b denote the axes major and minor of the ellipse, 



Z>^_ (DH-F)H-(D-F)sec2<^ 

 a« - (J) + F) - (D - F) sec2<^' 



