222 Professor Barlow on Achromatic Telescopes 



other, are dependent and proportional in each respective fo- 

 cus, if these proportions do not arise from the natural proper- 

 ties of the two media, they cannot be produced by art, while 

 the lenses are in contact ; but in any case where the violet side 

 of the correcting or concave lens exceeds that of the red in a 

 greater proportion than the violet exceeds the red in the con- 

 vex lens, and if the dispersive ratio be so great as to admit of 

 the lenses being sufficiently opened from each other, then such 

 a distance may be found as shall produce the above propor- 

 tion ; and hence unite these three rays in one common focus, 

 or at least approximate towards this result. 



" Let the distance of the lenses be d, and let the plate focus 

 remain as before/, then the negative focus must be reduced till 



~ "~", ■ = dispersion. * 



" Conceive this length to be found, which may still be de- 

 noted byy, and r 7 and v' may also denote the same as before, 



t' v 1 



the ratio -r, to -rr, — -, will likewise still remain the same. 



J — r J -H« 



" But the coloured focus of the plate lens remaining the 

 same as before, while the mean focus is changed fromyto 

 J'—d\ we must now, in order to unite the three rays, have 



/_ d + r ' f— a — v"f + r' ' f — »' 

 The two latter terms remaining in the same ratio, while the two 

 former may be varied at pleasure by changing the value of d ; 

 which will have no effect on the ratio of the latter terms, al- 

 though the actual value of/', r 1 and t/, must necessarily vary 

 for every change in the value of d. 



" Let the constant ratio of the latter terms be m : n ; then 

 we have to find d, such that 



v v r mv 



or, 



f—d+r '/— v — v ' J — d + r J—d—v 



,.. , , . , mv f 4- mv r — n r f A- n vr 



which reduced, gives d = — ! — ■ . 



° in — n r 



" If now a\ a", a"' be taken to denote the refractive indices 



of the red, green, and violet ray in the front lens, we shall find 



f 9 r and v, to be to each other as 1, -, and — ^ — ^and 



* a' a 



* Phil. Trams. 1827, A in. xv. 



