THE THEORY OF OPTICAL INSTRUMENTS. 



441 



The form of this equation differs from that for the primary focal lines 

 only by the omission of cos 2 6, and by substituting b for a. 



From these equations we obtain the following values of the cardinal points 

 of the ray in the primary and secondary plane : 



1 



Let 



A' = 



db? 



, B = 



i 



B' = 



= ' D= 



' TV 



= 5&7' = B'^-A'C' 



(8). 



Primary. 



Secondary. 



(9)- 



Value of r\ at 1st principal focus g l = ^ l ( 



Value of r., at 2nd principal focus g. 2 = ^AD cos 2 2 



1st principal focal length ^BD cos Q l cos # 2 



2nd principal focal length f 3 ^BD cos 1 cos 0. 2 f.{ = p. 2 l 



In tracing the axis of the pencil, we may, as a first approximation, neglect 

 the square of 6. 



Let it cross the axis before incidence at H 1} and after emergence at T 2 , 



where H, is -f t beyond the first principal focus, and H, is a/, beyond the 

 a 



second principal focus. 



Let it cut the first and second principal planes at a distance f from the 

 axis. Then, since these planes are at distances / and / 2 within the principal foci, 



fa 1 



l+a 



.(10), 

 .(11). 



Let the object and image be respectively - R f, and $ beyond the principal 



foci, 



V0 T fl/' 



< 12 >' 



X 1 = 



We shall next determine the relation between the curvature of the object 

 and that of the image. 



VOL. IL 



56 



