The Microscope. 209 



THE DIOPTRICAL PRINCIPLES OF THE MICROSCOPE. 



GEORGE MACLOSKIE, SC. D., LL. D,, 

 PROFESSOR OF BIOLOGY, COLLEGE OF NEW JERSEY. 



FOR calculating the path of light through a centred system of 

 lenses (omitting spherical and chromatic aberrations), the 

 formulas hitherto used are unnecessarily complex. I have tried 

 an easy formula given in Matthiesseii' s Dioptrik, and find that it 

 can be extened to all cases, so that a single formula, easily ob- 

 tained and easily remembered, can be used to determine the focal 

 lengths of lenses, doublets, objectives and of the entire optical 

 system of a microscope or telescope, and a slight variation of it 

 enables us to find the principal planes of lenses real and imagi- 

 nary. Thus we can understand our favorite optical instrument 

 without using the troublesome algorithm of continued fractions 

 given by Gauss. 



The method may be best illustrated by indicating its applica- 

 tion to the problem of an objective given in Ndgeli and Schwen- 

 dener^s book on " The Microscope.^^ It is required to find the focal 

 lengths of an imaginary lens equivalent to an objective consisting 

 of three doublets, each doublet being a plano-concave flint-glass 

 lens backed by an equiconvex crown-glass lens, the refractive 

 indices of flint, crown-glass and air being given. 



I. For the surface refractions. The refraction of a ray through 

 a surface from medium with index no to another with index ni, 

 produces two focal lengths, a first focal length, /i, for rays enter- 

 ing medium no, and a second gi, for rays entering medium ni ; 

 and the general equation is 



no r n-L r 



nx no 



no — 1 ni — 1 



ni no 



whereby, making r positive for rays approaching a convex sur- 

 face, and negative otherwise, we can find the surface refractions. 

 For the flat face of the flint lens we use the two values / = — 

 oo, g = 71 X CO, as the ratio of the two values is required. 



II. For the Lenses. Having got the surface refractions (fi, giy 

 for the first surface refraction, and/j, g^ for the second) we apply 



