Mr Stevenson^s Totally Beflecting Mirror. 147 



To apply these results to the case before us, making F the 

 origin, and B F the axis of a:, fig. 2 : 



The eo-ordinates of A are —xf^y'; and of B — ar" 0. 

 Therefore, in the preceding formulae, putting a:* =: — • aj', and 



j._ y ^ y 



cos d— cosy 2 sin 7 + ^ siny—^ 



a—'-xf + r sin ^ 



, y' cos y 



6 = — — ^ 1 — = — r cos 7 



cos d ~ cos y 



Also, putting 4r for the whole angle which the breadth of one 

 of the zones subtends at the point F, and d for the radius of 

 the mirror measured from F to the point A, the angle 



A F B = '^, a/ = fl? cos ^, and y'=d sin ^. Next, because the 

 ray A F is reflected in the direction A C perpendicular to 

 B F, C AF= 90 -"I ; and since the normal A D to the re- 

 flecting surface bisects the angle C AF, therefore C AD = 



45°-t 

 4 



Now, sinoe A and A D are respectively perpendicular to 



the tangent at A, and to the line B F, it follows that, the 



angle C A D is equal to the iacUaaiiion of the tangent 



to the line B F ; therefore 6 = 45V y 



Again, because the ray A B is reflected at B in a direction 

 at right angles to B F, the tangent at B is iBclined at aaof 

 angle of 45° to B F. Hence 7=45°. Therefore, substituting 

 these values of 6, 7, xf, and y\ we obtain finally 



d sm ^ 

 r = — 



2 sin (45°-^\ sin t 



I 



J, 

 0= — c? cos -^ + r sin 



6= — r sin 45° 



in (45=- 1) 



(1) 



e2 



