560 Mr potter, ON A NEW CORRECTION 



with 7', for a ray which passes near the edges, and are therefore neg- 



lisible ; and also 4^ — ^rr^ is again a much smaller quantity, and 



'' 1 . 2 . ((^i Ji, )- ^ 



therefore not needing attention : so that we shall consider /, and f, as 

 the thicknesses of the lenses af or near their edges, and as constants. 



Differentiating the expressions for D and D', and substituting in 

 the expression S (JD — D') = 0, we find, after the reductions, 



= ^ f>{\ • O/" + 7- -CM r- ■ - I 



P\ p-: I \ /" M" / r-,p.i IX- r^r,] 



f /m' - 1 5, « - 1 . A 1 m'" — 1 om' \ 



\\ !>■ H-' I r;,f>, M- r.rsi 



In the ordinary formula, the condition that a double object-glass 

 shall be achromatic, is 



^ _ V _ „ 



So that the expression we have obtained consists of the common one 

 together with other terms involving the thicknesses of the lenses. 



The other method which I have mentioned will be easily com- 

 prehended from Fig. 4. 



If abed in this figure represent a double object-glass, and R„ R,, R^, 

 Mi, M-i, M-i, 9,, q, qi, represent the same points as in the two first 

 figures; J?, R, R^q^ being the path of the ray incident at J?,, then we have, 

 tangent of the angle R^ q^ M,, in which the emergent ray meets the axis 

 at qi, 



_ R^M^ 



~ q,M,' 



_ ?A . 



q,M,' 



and, in order that the combination may be achromatic, we must have 

 the variation of this / R^qiM^ (or its tangent since it is small) = 0, 

 whilst M and n', the refractive indices, vary for the different colours of 

 the spectrum. For this purpose, expressing iji in terms of y, and the 

 radii of the surfaces, and distances R^Ri, RoR^, in the lenses; and q^M^ 



