OF TELESCOPES AND MICROSCOPES. 73 



the relative positions of three given lenses, which shall form a 

 combination achromatic for rays of any degree of obliquity. 



In the particular case in which the focal distances of all the 

 lenses are equal, and the intervals a^ 2 , &c. are also equal, the 

 general equations degenerate into a system of simultaneous equa- 

 tions in finite differences. They are then 



Eliminating z n , we get 



y*-(p a 

 The general solution of this will be 



A being a root of the recurring quadratic equation 



x z - (pa + 2) x + 1 = 0, 

 c and Cj are to be found by the conditions 



y 1 = cA + ^ ^l" 1 , y l + pz 1 = cA* + c, A"*. 



The general solution of the system of quasi-equations em- 

 ployed in the enquiry must involve some functional operation 

 which degenerates into the radical contained in A. 



It would be perhaps worth considering how far we might be 

 able to present this operation in a distinct form, defined and 

 distinguished by a particular symbol; but the subject is not 

 one which can be discussed at present. At any rate, we see 

 that the research of the general expression for y n is one of 

 considerable difficulty. 



The greater part of the investigation given by Mr Airy in 

 the conclusion of his paper, with respect to the achromatism 

 of microscopes, becomes unnecessary by employing the general 

 expression given above for^ 4 . His object is to determine the 

 distance of an object-glass of given focal length from a diaphragm 

 whose distance from the field-glass of a given eye-piece of three 

 lenses is given. 



Let a Q be the distance of the diaphragm from the field-glass ; 

 therefore we have ^ = a Q y^ and putting this value for z l9 

 we get an expression for y^ of the form y 4 = y v ~R. The chro- 

 matic variation of this is to be zero, and consequently that of 

 its logarithm ; 



