66 BUIiLETIN OP THE 



consisting of infinitely thin lenses in contact, in order that, with 

 any given law of dispersion whatever, the greatest possible 

 number of light-rays of different degrees of refrangibility may be 

 brought to a common focus. 



For any system of thin lenses in contact we have 



y = 0.,— 1)A, + (^,-l)A,+ (^3 — l)A3 + etc., (1) 



the number of terras being unlimited. For a dispersion formula 

 we write 



^ = tW. (2) 



The form of ^ (%) is unknown, but there will be no loss of gener- 

 ality if it is developed in a series arranged according to the 

 powers of %. We, therefore, have 



^ = a-|-6^'"-|-c^"-f-e?iP + etc., (3) 



in which a, b, c, etc., are constants, and the number of terms may 

 be taken as great as is desired. 

 Let us also put 



C = A,(a, — 1)+A,(a,-1)+A3(a3 — l)-i-etc. 



D = Afi, -f A,b, -f A3 &3 + etc. (4) 



E = AjCj -f A3C2 -f A3C3 + etc. 



F = Aiei 4- A 362 4- Aggg -f- etc. 

 etc. etc. etc., 



the number of these equations, and the number of terms in the 

 right hand member of each of them, being the same as the num- 

 ber of terms in the right hand member of (3). Now substituting 

 for the ixs in (1) their values in terms of the auxiliaries C, J), E, 

 etc., of the equations (4), we find 



— = C+D7i'°-f-Ex" + FxP+etc. (5) 



Considering ■>, as the abscissa, and f as the ordinate, this is 

 the equation of the focal curve. Its first derivative, with respect 

 to / and X, is 



df 



~ = —f' (mDx""-^ + nEx°-^ + etc.), (6) 



which, as is well known, expresses for every point of the curve 

 tiie tangent of the angle made by the tangent line with the axis 

 of abscissas. The number of rays of different degrees of refran- 

 gibility which can be brought to a common focus will evidently 



