﻿346 Mr. F. L. 0. Wadsworth on 



to the cross-wire of the observing telescope. On examination 

 of the geometrical path of the ray through the system just 

 indicated, it was unexpectedly found that if two simple con- 

 ditions were fulfilled, the refracted ray was, after reflexion, 

 not only constant in direction but also constant in position, 

 viz., that there was no lateral displacement for different wave- 

 lengths, and consequently no angular correction required for 

 either plane or spherical waves. To find these necessary 

 conditions, let us first consider the general case in which the 

 axis of rotation and the reflecting-mirror are in any position 

 with reference to the line of collimation. 



Let the axis of collimation be the X axis, and let a line 

 passing through the axis of rotation E, be the Y axis. Let d 

 be the perpendicular distance from the axis of rotation to the 

 plane of the mirror-face ; let b be the perpendicular distance 

 from the same point to the plane bisecting the angle of the 

 prism (and hence bisecting also the angle of deviation of the 

 ray at minimum deviation), and let a be the y coordinate 

 of this point. Then the equation of the line E F (the inter- 

 section of the plane of the reflecting surface with the X Y 

 plane) will be 



d 4- a cos a 



y= tan a a; ; 



° cos a 



and of the line A B (refracted ray), 



y tan X +^^{ sin 0/2 + 6}. 



The coordinates of the points of intersection of these two lines 

 will therefore be 



