MHMOIKS OV TllH NATIONAL ACADK.MV OF SCI liNCKS. ;{vj 



irthc lenses are all imk'tinitcly thin and in ronlait— in otlier words, if all tlie. t'n art- equal U, 

 ztMd, and therofore all tlio ;(\s f(|ual to unity, tin- e<|uations («) can he readily «(imbined into one. 

 Tliis rodiution, by inspeition, is 



c.,,=;'Mi-^J^)+r* '(1-^^-')// + +y(\-,j)(fj^fj^-i tj']+ ,j^ ,■'. 



If «„ is tlie index of rrliacliun of tiie lirst rnedinin, «' in tlia( of tin- second, and so on. we 

 shall have 



and the product of all these quantities will be equal to "" ,, \vhi< li qnantity, the ratio of li'dit 



veloeity in the last medium to that in the first, wo will designate by ii,.. I'.y tiiis sni)stitntion the 

 above equation becomes: 



C, =1' + po<-S 

 where P is a constant, depending solely upon the physical constants of the system ; it is called 

 X\\o power of the combiuaton, not only because it is the change in curvature which the system can 

 produce in a plane incident wave surface \^c' =0), but also because in the most common of all cases 

 in which the lirst and last mediums are alike, air, for example, and therefore Pq = 1. 1* is the 

 curvature which the system adds to every incident wave surface. 



II — IMAGES AND MAC.NIFICATION. 



Fig 2 



Let o' be the iudetinitely small distance of the center of the incident wave surface c' from the 

 axis and <>, that of the center of the refracted wave suiface Cj; then we have, from the general 

 law of wave motion, the line Oi is the image of o' and the angle which c, makes with y at the vertex 

 is ij times that which c' makes at the same point, hence 



The image of Oi after refraction at a second surface we may call o^, then by similar rea.soning 

 we have 



Extending this jirocess to z.+ l refractions and ((illecting we have 



,--=p^-> 



^•=pA 





