74 Dr Anderson on Curves generated by 



makes with the axis of rotation. Hence the apex of the se- 

 condary curve, or the point where that curve intersects the 

 axis, depends solely upon the inclination of the mirrors ; but 

 the position of the node varies with the direction of the light. 



Having investigated the more important features in the 

 forms of the curves, generated by double reflection, we shall 

 now determine the conditions upon which their formation de- 

 pends. For this purpose we must evidently assume that, in 

 the formula (R), we have cos i—o^ or that the angle of inci- 

 dence for the second reflection is ^bout to vanish ; we thus 

 have — 



Cos r=2 cos 2 /3 cos I. 

 Or, 

 Cos « sin A sin |8 -f-cos A cos |Sz:2 cos 2 ^ ( — cos et> sin A sin /S-f cos A cos /3). 



XT n 2 cos 2 y3 - 1 ,^,. 



Hence, Cos a = tttt, ^tttt — TT — ~» . . . . (<J ). 



' (1+2 cos 2 /S) tan A tan /8 ^ ' 



This expression may be greatly simplified by various substi- 

 tutions and reductions. 



In the first place, we have — 



Cos a tan A (1 + 2 cos 2 /3) tan /3 z: 2 cos 2 /3 — 1. 



Or, Cos ct tan A (1+2 cos 2^) (^tl_££L?^ V^- = _ (l -2 cos 2^). 



\1+ cos 2/3/ 



^ „ , „ 1+4 cos3 2 ^ - 3 cos 2 i8 

 Cos2 ot, tan2 A = 



Cos 2 « tan2 A 



1 — 4 cos3 2 ^ + 3 cos 2 /3 

 1+ cos 6 /3 cos2 3 /3 



1— cos 6 ^ sin2 3 /8' 

 Hence, Cos a, tan A = cot 3 /3. 



And Cos «e =r cot A cot 3 /3, . . . . . ; (H'). 



When cos a = l, or (t=.Oy we obtain — 



Tan A r= cot 3 /S, . (F). 



Hence, the condition of there being no second reflection, 

 when a=o, is, that X shall be equal to the complement of 3 /3, 

 or less. Since the greatest value 3 /3 can have is 90°, /3 cannot 

 exceed 30°, in order that there may be no secondary curve.* 



* It might also have been assumed in the formula (H') that « = 90° ; 

 and we should thus have obtained cot a cot 3 jS = a, a result which im- 



