for a double Telescope Objective. 293 



dn <Pri 

 6( P=-^ A dn' drf Sn > W 



which is the measure for the secondary chromatic abberration 

 for a binary lens of materials n and n' and power <p. Although 

 it is, as appears from the method in which it was derived, an 

 approximation, it is found to be perfectly accurate to three sig- 

 nificant figures from the wave-lengths A to H inclusive. 



If we already have the constants of the equation expressing 

 the relation between n' and n the calculation of d<p is easy and 

 the method obvious. For the Crown 1219 and Flint 1237 of 

 my paper on double objectives cited above, which may be 

 taken as typical specimens of the crown and flint glasses 

 used in the construction of astronomical telescopes during 

 the last half century, the value of dcp is —27*1; but if we have 

 not sufficient data to enable us to calculate the values of a, /3 

 and y, or if we wish to avoid the labor of determining them, 

 we may content ourselves with three indices for each land of 

 glass employed and be confident of a useful approximation to 

 the true solution if the corresponding wave-lengths are well 

 distributed through the spectrum. Thus if the indices for the 

 Fraunhofer lines C, D and F, are given, we may substitute in 



the above equations, Uv, for n . -/■ — y- for -r-f, and for ^— =2r 

 * D °' n' F —n' c dnl dn 2 ' 



the value deduced from the equations : 



n' F —n' D = (n F —tiv)fi + (%■—%,) V 

 n'-D—n'c = (?i D — n c )/3 + (n D .—n c yy. 



Such a ready approximation gives d<p= — 30*0 for the glasses 

 above mentioned. 



In the catalogue of the Jena manufacturers are given the 

 approximate indices of refraction for the line D, and to a much 

 greater degree of precision, the differences of the indices for 

 various intervals in the spectrum including the intervals C to 

 F and D to F. Since the color characteristics of a combina- 

 tion depend upon the differences far more decidedly than upon 

 the absolute value of the indices, as has already been stated, 

 we have in this list all the data necessary to secure an approxi- 

 mate solution to the problem by the method described. We 

 may most conveniently proceed by selecting some one from the 

 list and then find the value of o<p for each combination of the 

 others with it for a definite change of wave-length of light ; 

 or, what is far better for our end in view, calculate the value of 

 the coefficient of dn 2 in the expression for d<p. If we call this 



dn° d"n' 

 coefficient h its value will be — AA -=—r -r-r. Such tabulated 



an,, dn 



values of h would give us a notion whether a combination was 

 subject to a large secondary chromatic aberration or to a small 



