ON THE ACHROMATISM OF EYE-PIECES OF 

 TELESCOPES AND MICROSCOPES*. 



MR AIRY, in a paper published in the Second Volume of the 

 Cambridge Transactions, has investigated the conditions under 

 which a system of lenses is achromatic, per se that is, when 

 only one kind of glass is made use of. The enquiry, on account 

 of its immediate application to the eye-pieces of telescopes and 

 microscopes, is one of considerable importance ; and as the way 

 in which it is conducted in the paper just mentioned appears to 

 be more complicated than necessary, and does not lead to the 

 most general solution of the problem, perhaps the following 

 attempt may be not wholly without interest. 



The difficulty of the question consists in finding the angle 

 which a ray of light makes with the axis of the system of lenses 

 after having passed through it. When this is done, we have 

 only to take the chromatic variation, and equate it to zero, to 

 get the general equation of achromatism for the system. 



In what follows, lines are considered positive when measured 

 in a direction opposite to that of the incident ray. 



Let y n be the tangent of the angle which the ray makes with 

 the axis before its incidence on the n th lens of the system ; let 

 z n be the distance from the axis of the point where it impinges 

 on that lens; and take a n _ t to signify the distance of the n ih 



from the n 1 th lens. Then and are the distances of 



the conjugate foci from the n ih lens, and by the ordinary 

 formula 



p being the reciprocal of the focal length ; therefore 



# + l-#=^n- 

 * Cambridge Mathematical Journal, No. VI. Vol. I. p. 269, May, 1839. 



