DETERMINATION OF CORRESPONDING IMAGE-POINTS. 25 



angle 8 must be taken instead of the . tangent. With the usual 

 methods of observation this does not make much difference ; for 

 the sine of 15, for instance, is not as much as "01 of the radius 

 smaller than the tangent of that angle ; whence it follows, among 

 other things, that the small image of the diaphragm-aperture in 

 the posterior focal plane of the objective appears, according to 

 both formula?, of nearly the same size, the incident pencil of light 

 assumed to be 30 aperture. With an objective of 3 mm. focal 



2 x tan 15 



length, the diameter, d, in the one case = . 3, or 



r 



2 x sin 15 



1*60 mm., and in the other - . 3, or 1*55 mm. The 



r 



difference amounts, therefore, only to ^ mm. But instances are 

 found in practice, where the differences between sine and tangent 

 are much greater, and where it is not permissible to disregard 

 them. With reference to this we give the following theorem put 

 forward by Abbe, 1 which is applicable to all angles that are met 

 with in microscopic observation. It is " Wlicn an optical system 

 is completely aplanatic for one of its focal points, every ray emerging 

 from this point meets a plane drawn, through the other focal point 

 cut a distance from the axis, the linear magnitude of ivhich is equal to 

 the product of the equivalent focal length of the system and the sine of 

 the angle which this ray makes with tJie axis" 



The proof of this proposition has not yet been published by 

 Abbe. It will suffice, however, for our purpose to show that with 

 any double-lens the tangent of the angle in question must, in fact, 

 give somewhat too high values for the distance of the emergent rays 

 from the axis, that is, if we determine the cardinal points for 

 such a lens, it will be found that the anterior principal plane will 

 lie somewhat behind the geometrical centre of the crown-glass lens, 

 and will, consequently, intersect the last refracting surface. Com- 

 pare, for instance, the calculations given on the succeeding pages, 

 or the construction of Fig. 7. If we suppose a ray proceeding 

 from the focal point and passing without deviation to the point 

 of intersection of the anterior principal plane with the last refract- 

 ing surface, it necessarily meets this surface at a greater distance 

 than if it were diverted inwards at the anterior surface of the lens, 

 and then assumed, almost in the same vertical, a direction parallel 

 to the axis. The differences that result from this may be shown, 

 1 " Archiv fur mikr. Anat." Bd. ix. p. 420. 



