of Stereoscopic Phenomenon. 331 



3. Before the further consideration of this experiment and its 

 consequences, the following short analytical introduction may be 

 permitted. 



Let the axis of rotation be the axis of z, the plane in which 

 the cylinder rotates be the plane xy. Let the eye A be at the 

 distance r from the origin ; the line OA making the angles 

 u, (S, y with the axes of x, y, and z respectively. And let the 

 angles made by the parallel incident rays with the same axes be 

 called u v j3v and y x . The axes of x and y may, however, be so taken 

 that the axis of y is in the plane passing through the axis of z and 

 the ray of light incident on 0, so that 0^ = 90°, l = 9O°— y v 

 Let the angle formed at any moment by the cylinder (which 

 must be regarded as infinitely thin) with the axis of x be called 

 <f>. We have now to inquire what is the distance of the point 

 of reflexion 11 from the origin O. Let the distance in question 

 be called p. In the first place, the angle ROA is easily deter- 

 mined as follows : 



cos ROA = cos <f> cos a + sin <f> cos j3. 



The angle ARO made by the reflected ray with the cylinder 

 must, from the property of the cylindrical surface, be equal to 

 that made by the incident ray, whence 



cos ARO = sin y l sin cj> 



In the triangle AOR we know, therefore, the angles ROA and 

 ARO, and the side AO = r. Whence we get 



_r sin (ARO 4- AOR) 

 AK ~P- ^sinlARO) ; 



which, on substituting the values of ARO and AOR, becomes 

 cos <fy cos a. + sin <p cos /3 + sin y Y sin <f> 



/] — (cos <f cos a + sin $> cos /3) 2 ~| 

 V 1 — sin 2 7j sin 2 <f> J 



a curve of the sixth order, closed, unless 7^90°, and passing 

 through the origin. 



When the incident rays are parallel to the axis of rotation, 

 y x = 0, and the part of the above expression containing the square 

 root disappears. The equation being then referred to rectan- 

 gular coordinates, becomes 



/ r \ 2 / r \ 2 r 2 



[x-^cosuj +(^/-~cos£j =-sin 2 7, 



that is to say, the equation of a circle having the projection of 

 OA for a diameter, and being the intersection of the plane xy 



Z2 



=r\ ( 



