+/ 



4 . 



46 



there remains 



Mr. Mac Cullagh on the Laws of 



sin^2t /m 



sm''(t-j-Otan''0 — sin2(t — ()tan'0 = „-) 



Vila' I V 1 2/ 3 cos^02 



and this, by making 



(III 



sin2t„ 



sin2( 



'COS 



'e); (9) 



m. 



sin2( 



becomes 



sin'(t,+tjtan'e,— sin^(«,— tJtan'6'3=sin2/,(sin2«^+2/t)tan'0^, 



which is divisible by equation (8), the quotient being 



sin(,,+<jtan0,+sin(t,— t^)tan6i3=(sin2<^+2A)tan0^. 



Then, by adding and subtracting equations (8) and (12), we obtain 

 tane =cos(t,— i )tan6> + . , , . , 



' V 1 2' 2 ' sin((,+( ) 



(10) 



(10 



(12) 



tane. 



/ . \ ., /«tan0 

 -cos(i,+(Jtanft 4- -^- -. 



(13) 



These equations give the positions of the incident and reflected transversals when 

 h is known. 



Now let the directions in which the transversals have been resolved in 

 equations (2), (3), (4), be taken for the axes of z, x, y respectively ; so 

 that, the origin being at O, the plane of xy may be the plane of incidence, and 

 the axis of x may lie in the surface of the crystal. And, the reflected ray 

 being conceived to lie within the angle made by the positive directions of x and^y, 

 let the initial condition that we have assumed for the angles 6^, 6^, 6^ be satisfied 

 by supposing that, when these angles begin, the transversals t,, t^ lie between the 

 negative directions of x and y, and the tranversal r^ between the directions of 

 -\- X and —y. Then if 6*,, 6^, 6^ be reckoned towards the positive axis of z, so 

 that each angle may be 90° when the corresponding transversal points in the 

 direction of z positive, the equations of the transversal t^ will be 



(14) 



tanfl, 



cost, 



suit, 





^ J ^ rf. yi/a. 



O^tiH //^^^»*^ p 



y 



/y 



'0 





