1905.] The Theory of Symmetrical Optical Objectives. 397 



(2) the plane of the stop, (3) and (4) its images with respect to the 

 front arid back components, and (1) the plane symmetrical to (5) ; Pand 

 Q are two points on the stop equidistant from the centre, a and b two- 

 parallel rays through these points, c and d another pair of parallel rays 

 through these points ; the intersections of these rays with the planes 

 1, 3, 4, and 5, being a l5 a 3 , 4 , 5 , ^, etc. The ray d is chosen so that 

 b,b 5 is parallel to dd 5 , then by symmetry it is evident that CiC 3 is. 

 parallel to aia s . It is also evident that the planes 3 and 4, being 

 images of 2, are in every way similar ; hence they are the principal 

 planes of the whole system, and the focal plane of the combined 

 system is mid-way between 4 and 5. Let the various rays intersect 

 this plane in A, B, C, and D. 



The curvature error* of the combined system can be measured by 

 CA, less the effects due to spherical aberration, where CA evidently 



= J (C 4 4 ~ 



but, since 6 4 6 5 is parallel to 

 CA = | 



Thus omitting effects of spherical aberration in both cases, it is 

 evident that the curvature error for an angle o> and aperture angle 

 2a is the mean of the curvature errors for the single system for the 

 angles (w + Ja),and ((u --|a), where (o-w represents the angular 

 value of the distortion of the single lens, together with the portion of 

 \ (^4 - & 4 ^ 4 ), which is not common to all angles. 



When the meridianal astigmatic curve is drawn as usual the focal 

 lengths being unreduced and the abscissae representing angular field 

 the ordinates for the combined system will be one-half of those for 

 the single system, subject to the corrections from terms corresponding 



To appreciate the value of the latter expression draw through P the 

 rays a, d, symmetrical to a and d with respect to the axis, then 6 4 c 4 

 4 ^ 4 = c 4 a' 4 - e 4 d' 4 , and since the angles between b and a, and c and d, 

 are equal, this quantity will, in general, be small when the stop and 

 its image are close together. 



To examine the spherical aberration let c 4 5 be parallel to the axis, 

 a 4 5 also parallel to the axis, then d\d^ and b^ are also parallel to the 

 axis (fig. 2). The spherical aberration of the whole system is 



but c 4 4 



therefore DB = J {c 5 d 5 + a b h - (c 4 5 4 + a 



* This term is used for the defect due to the parallel rays through P and Q 

 not intersecting on the focal plane, the effects of spherical aberration being allowed 

 for. 



