Curvature Method of Teaching Ojttics. 485 



Hence, for tracing the course of a pencil through any 

 system, we have simply 



Surface 0. try — o'-i = JV>, ° ] " er 1 / =F o 74 O +0''-] ; 

 Interval 1. fi 2 — /*o=°"iV or n 2 = t \°'i+ n o', 

 Surface '1. rr 3 ' — rr'_i = F 2 A 2 or cr 3 ' = Fo// 2 + o~\ ; 



and this may be extended to any number of surfaces by 

 putting down the convergences and reduced thicknesses, and 

 the corresponding lateral intersections and reduced angles in 

 the order 



F h' F 2 *,' Ac. 

 a' -i h &i h 2 a.J &c. 



each member of the lower series being derived by multiplying 

 the penultimate member by the corresponding value above 

 it, and adding the antepenultimate. This is the extremely 

 valuable method due to Von Seidel *. 



For a single thick lens we have to take simply the three 

 equations given, and we find 



h 3 =t l , a ] ! + h =(F t l ' + 1)^ + *! V_ 1 =CA + D<r'.-i 



and 



a 3 > = F 2 h 2 + a 1 ' = (F 4 F 2 + F F 2 ^)A + (F 2 ^ + iy_ 1 = Ah + Ba'.,. 



We may express this either in the ordinary Gauss or in 

 the convergence form by taking 



v .: = h y or V/=^. 

 In the latter case 



Vq=7T7-^ — ft-/ — = ^ utt/ as before. 



Extension of Convergence Theory to any System. 

 Our equations for the convergence method were 



Vi'-U'-i^F.; 

 IT' X • 



v ; :-u 1 '=f 3 , &c. 



* Dr. 8. P. Thompson's translation of Lnmmer'e Photographic Optics, 



Appendix If. 



