JAMIN ON METALLIC REFLEXION. 



87 



in the direction of the principal section, {y') that in the direction 

 perpendicular, 



a/ = y sin eo + x cos w, 



y' = y cos CO — jr sin <o. 

 These two vibrations may be written 



0?' = A'cos(2 7r^+8'V 



y'= B'cos(2^p^ + 8"V 



and A', B', 8', 8" will be obtained according to Fresnel's rule. 

 These quantities will be 



A'^ = sin^ a sin^ w + cos'^ a. cos^ w 



+ - sin 2 « sin 2 w cos 8 . . . . vibration in axis of <«■, 

 B'^ = sin- a cos^ w + cos^ a sin^ a> 



— 2 sin 2 a sin 2 CO cos 8 . . . . vibration in axis of y 



, (13.) 



tan 8' = 

 tan 8" = 



sin a sin so sin 8 



cos a cos CO + sin a. sin co cos 8 

 sin a cos oi sin 8 



— sin ui cos a + sin a cos co cos 8* 



These latter formulae serve to calculate the difference of phase of 

 the two rays ; they give 



, /sji sj»s sin 8 sin 2 « 



tan (8' — 8") =-^— , . ^4 \ 



sm 2 CO cos 2 cz — sm 2 a cos 2 CO cos 8 ^ ' 

 If we wish to find the direction for which the images are 

 maxima and minima, we must differentiate formulee (13.j with 

 regard to (co) ; they give 



— cos 2 a sin 2 CO + sin 2 a cos 2 co cos 8, 

 cos 2 c< sin 2 CO — sin 2 « cos 2 co cos 8. 

 These two differentials being equal, but of contrary sign, we 

 conclude that one of the images is a maximum when the other 

 is a minimum, and vice versa : and this will hold for the direc- 

 tion found by putting the differentials = zero ; which gives 



tan 2 CO = tan 2 a. cos 8, 

 a relation identical with that which gives the direction of the 

 axes of the ellipse. Therefore, 



