64f.v Dr Anderson on Curves generated by 



Cos r + cos 1 = 2 cos A cos /8 



Cos r — cos I =: 2 cos <« sin A sin /S 



} (L) 



TT cos I'i— cos I 



■tlence, z:zv~i ? = cos « tan a tan /3. 



cos I 4- cos I '^ 



And tan ^ (I -f F) tan I (F^I) = cos « tan a tan /3, . (M) 



And when a = /3 = 45", 



Tan \ (I + r) tan i (I - 1.) = cos «, (N) 



The condition of the angle of incidence vanishing, or that 

 of the incident ray being reflected directly backward, is, 1 = 0, 

 or cos 1 = 1. Hence, in that case, we have by the general 

 formula — 



Cos (t — cosec A cosec /S^cot A cot /3, (0) 



The limit of reflection takes place when the direction of the 

 luminous ray coincides with the plane of the mirror ; in which 

 case, cos I = cos 90° = o. 



And cos «fr = — cot A cot /3 ^ 

 Or sec «=^tanA tan/3 J 



Hence the limit of reflection cannot take place when the 

 angle of rotation is in the first or fourth quadrant. 



When the direction of the luminous ray coincides with the 

 axis of rotation X=o, and the general formula becomes — 

 Cos I = cos /3, or I n /3. 



Hence the reflected ray describes, around the axis of rota- 

 tion, the convex surface of a cone ; and, consequently, the 

 curves which it generates are the various sections of that solid, 

 according to the position of the planes on which the images 

 are received. 



Having thus derived a considerable number of formulae for 

 determining the angle of incidence in the case of a single re • 

 flection from either mirror, we shall now proceed to the in- 

 vestigation of a more complex case, namely, that of deducing 

 an expression for the angle of incidence, when the ray, after 

 being reflected from the surface of one of the mii-rors, suffers 

 a second reflection from the surface of the other, against which 

 it impinges in certain positions of the revolving mirrors. For 

 this purpose, we shall find, in the first place, an expression for 

 the variable angle M'L N', formed by the planes of reflection, 

 when the pole of one mirror is at M', and that of the other at 



