294 C. S. Hastings — Secondary Chromatic Aberration 



one, but the numbers would not be proportional to the aberra- 

 tions because of the variable factors dn* ; but if we multiply 



/ dn \ 2 

 the number k by ( — ) , where n is the index for the glass 



which is to be combined with that common to the whole series 

 of pairs, and n x is the index for another glass arbitrarily as- 

 sumed to have the standard dispersion, we shall have a series 

 of numbers all of which are to be multiplied by the same 

 quantity, namely, dn*, to find the secondary chromatic aberra- 

 tion for a binary objective of each pair. These numbers we 

 will designate by h'. In order to make the comparison with 

 an objective of the ordinary type, I have chosen No. 37 of 

 Schott and Co.'s catalogue, an ordinary flint of the kind gen- 

 erally employed in telescope objectives, for the negative lens of 

 all the combinations, and No. 13, which is an ordinary crown 

 glass, as the material of standard dispersive power. The fol- 

 lowing table contains the results of the computations thus in- 

 dicated. The first column contains the catalogue number of 

 the glass combined with No. 37;. the second column contains 

 A, the sum of the curvatures of the two surfaces of the posi- 

 tive lens, and the third, under B, the sum of the curvatures of 

 the two surfaces of the flint lens ; finally, the fourth column 

 contains k', which may be regarded as the true measure of the 

 secondary chromatic aberration. In short, if r 1 and r 3 are the 

 radii of the positive lens, r\ and r\ those of the flint No. 37 

 lens, we have for a binary lens of focal length unity, 



1 1 _, 1 1 . , 7 ' / dn 



A=- + -, B = -r + -r, k'-. ' 



'\dnj 



Inspection of the table shows — First: that only one of the 

 binary combinations having "ordinary dense flint" as one com- 

 ponent is practically free from secondary chromatic aberration, 

 namely, No. 30. This is described in the catalogue as a silicate 

 flint with relatively high refractive power. Although the com- 

 bination demands rather deep curves for the lenses, it is in my 

 opinion by far the best which the present resources of practical 

 optics affords, and is sensibly perfect. The inconvenience of 

 excessive curvature could be reduced by making the objective 

 of three lenses, the flint 37 being a double concave between 

 two positive lenses of flint 30, or, perhaps better, in the case of 

 large telescopes, increasing the ratio of focal length to aperture. 

 This conclusion stands or falls with the accuracy of the data 

 supplied by the catalogue, for, although there are strong reasons 

 for supposing the data good, I h^ve not seen either of the 

 materials in question. 



Second : that there are only two combinations for which k' is 

 positive, those of 66, an " ordinary light flint," and rock salt. 

 The first of these is of no practical interest on account of the 



