100 THEORY OF THE MICROSCOPE. 



their combined action is to be harmonious. For instance, if the 

 angular aperture of the Microscope is such that the delineation of 

 fine lines, whose interspaces are 1 mic., is just possible, it would 

 evidently be a fruitless labour to increase the amplification much 

 beyond the amount which experience has shown to be sufficient 

 for their observation. Similarly, the selection of too low a magnifi- 

 cation must be regarded as a mistake, since then the surplus of 

 resolving power, which is given by the angle of aperture, is 

 completely lost to the eye. An improperly chosen ratio between 

 the focal length and the angular aperture involves, therefore, in 

 the one case, a melcss magnifying power, not represented by any- 

 thing in the image ; in the other, a useless differentiating poiver, 

 which resolves more details than can be seen. 



The practically best proportion between the two factors cannot 

 be determined, once for all, with mathematical precision, on account 

 of the differences of construction. A table of corresponding mag- 

 nitudes cannot, therefore, be given here, arid we must confine 

 ourselves to the explanation of an example of the points which, 

 under certain conditions, give rules for construction. Tor this 

 purpose we assume, as Abbe 1 states, that the angle of aperture of a 

 dry system should not exceed 110. The limit of differentiation is 

 thus arithmetically determined, and the distance d of the smallest 

 details, whose delineation is just possible with extremely oblique 

 illumination, amounts 2 to %3 mic. In order to render such details 

 visible, it is necessary to magnify them to about 120 mic., and in 

 order to observe them conveniently, 180 or even 210 inic. would 

 not be too high. 3 Consequently, we get for the case of simple 



1 Of the absolute accuracy of this angle we do not feel convinced. It is 

 based upon the supposition that the curvature of the lens- surf aces is exactly 

 spherical. We are, however, informed hy practical opticians that it is possible, 

 in the polishing of a /one of the lens between the centre and the periphery, to 

 give a somewhat less decided curvature than the exact spherical form, by 

 which an enlargement of the angle of aperture would be feasible. 



2 The angle of the illuminating rays was here taken at 80 (generally too 

 high), and half the angle of aperture equals 55. We therefore get the equa- 

 tion 



wave-length 



d- 



sin 80 -f- sin oo c 



r> These numbers vary according to the sharpness of outline of the image. 

 With the naked eye, the divisions of an eye-piece micrometer, even if the 

 interspace is *1 mm., may be perceived with favourable illumination; but a 



