292 C. S. Hastings — Secondary Chromatic Aberration 



the determination of eight or ten indices of refraction for 

 known wave-lengths as well as a protracted calculation. The 

 manufacturers supply in their catalogue, however, the approxi- 

 mate refractive indices of their materials for several known 

 wave-lengths, and the differential refractions with considerable 

 accuracy. As the character of the color correction of an ob- 

 jective depends upon the latter quantity, it seemed that some 

 method might be devised for a systematic study of all the 

 materials founded upon these given constants, and thus avoid 

 the large amount of labor required in the old method. This 

 consideration led to the following solution. 



The power of a binary lens having the sum of the recipro- 

 cals of the radii of the two lenses respectively A and B is 



q)={n— l)A + (/i'— 1)3, 

 which, for a definite value of n, we will arbitrarily assume to 

 be unity, thus : 



<Po=K-l)A + K-l)B=l. (1) 



For an achromatic combination we must have 



^=0=A + ^B.-.B = -A^ (2). 



an an an x ' 



The value of the differential coefficient in this equation is a 

 variable, but the value which should be taken for the best color 

 correction for visual purposes has been shown to be* that cor- 

 responding to a wave length of about 5164. By employing 

 these values of n and n' in the above equations and designating 

 them by n e and n\, equation (1) becomes the condition of defi- 

 nite power and (2) that of achromatism. 



We now wish to find the expression for the secondary chro- 

 matic aberration. We may write 



cp =(n -l)A-(n' -l)^A; 



di\ 

 q) n =(?i + Sn — \) A—(n' +dn'—l)-j-^ A; whence 



<p e —q> n =6(p = (dn—r^ r 6n')A. 



But in the paper of vol. xviii, cited above, it was shown that 

 the refractive index of one species of glass can be expressed as 

 a simple trinomial function of that of another with practically 

 absolute accuracy, whence 



n' = a + fin -(- yn* 



dn'=(/3 + 2yn)dn-{-y3ri 2 ; also 



dn' dn' . . , d?n' 



e +2 r n =ch; = in: nearly ' and '=**?■ 



Substituting the value of dn' in the expression for dip, we have 



* This Journal, vol. xxiii, p. 167. 



