480 Dr. C. Y. Drysdale on the 



Combining these equations Ave have for any system 



+ +h' + F + U'_ 1 



V — F 4- 



hn—\ + F 2)l _ 



corresponding to the ordinary continued fraction of Gauss. 



We could derive the properties of complete lens systems 

 by the theory of continued fractions as was done by Gauss ; 

 but it is much more simple and satisfactory to have recourse 

 to a proof by induction, as the writer believes was first done 

 by Dr. Sumpner, for the ordinary Gauss method. 



Let it be granted that for a given system the equations 



A + B U'. 

 C+-DU'. 



1 

 m 2n -\ 



A' + BU'-i 

 U' 



AC 

 BD 





v 2n-\ 



hold. 



Now let the light pass through a space ^ n+1 and another 

 surface of convergence F 2 ». We shall have 



We then have 



_A + BU'_ 1 

 " U + DWZ' 9 



V 2n— it 2)1—1 + 1 

 V 2 n+1 = U' 2 „_i + F 2w . 



v: 



u. 



_ A(F 2 . W ^_ T + 1) + CF 3n + {BCF^g,.! + l) + DF 2 „} VL X 

 A4»-i + (B^_ 1 4-D)U'_ 1 



A' + B'CT. 

 C'+D'U'. 



where 



AC 

 BD 





A' C i , A(F 2 „C_ 1 + 1) + CF 2)l A*' 2w -i + C 

 B' D' J • BCF^W,! + 1) 4- DF 2 „ B* '*_, + D 



by the ordiuary properties of determinants. 

 Also 



1 1 1 



1 = A + BU'_ 1 



1 _ VWi.A' + B'U',, A^- 1 + C4-(B^_ 1 4-D)U^ 1 



™2n : U ; 2re _! C'-hD'tJ'.! A + BU'.! 



= A'4-B'U , _ 1 

 A+BU'_! ' 



