66 



Dr Anderson on Curves generated by 



Co8 i = (1 + 2 cos 2 |S) cos «. sin A sin ^4" (1 — ^ cos 2 /S) cos A cos /3 (S) 



In this form, th,e value of cos ^, though apparently a good 

 deal involved, becomes exceedingly simple when certain values 

 are assigned to X and /3. Thus, if /S=30°, in which case the 

 mirrors are inclined to each other, at an angle of 120°, we have 

 (1 + 2 cos 2 iS) = 2 ; and (1 - 2 cos 2 jS) = o. Hence, the second 

 member of the formula disappears, and 



Cos i = cos <x sin A , . . , . (T) 



Hence, when a=o, we have cos z= sin X ; and i is the com- 

 plement of X. This result may be illustrated by a geometrical 

 construction. 



Let A C be the axis of rotation, then B B' being the plane of 

 rotation, the angles B A M, B'A N are each 30°. Draw A'C 



parallel to A C, and let L A' the direction of the light, make 

 with A'C an angle L A'C, equal to 40°, and the angle L A'M 

 being 20°, the reflected ray A'D makes the angle A A'D also 

 20°. Hence, since angle D A A' is 120°, the angle AD A' is 40°. 

 But angle ADA' being the complement of the angle of inci- 

 dence, that angle itself is 50°, being the complement of the 

 original angle, which the direction of the light makes with 

 the axis. Or using X for ^^ C'A'L ; .^ L A'M = ^-^ 

 A A'D =60°- x; Hence ^ NDA' = 120°+ 60- x=180-x. 

 And ..^^ A D A' = X, the complement of the angle of inci- 

 dence A'D F. 



Let it be next supposed, that in the formula (S) we have 

 /5=45°, and the planes of the mirrors being at right angles to 



