the Spherical Aberration in a Lens-System. 85 



The expansion therefore takes the form 



#-i 



[{-lH©©Jf)© ^-©©>i©J))©} 

 sM'-K'-sH'-vXS'-K'-DO-I-'ei)' 



+3 ©')©} 



»©•-«©)©'}] 



a similar expression in ^u;, A;, and .<?/. 



To a first approximation neglecting (*^j and A we get 



of course 



Let each of these be denoted by Q S{ . 

 To a second approximation 



Whence 





fiiAj ^-iA M _ 1 



Si " S% — 





And if 6i be the inclination of the ray to the axis in the 

 medium whose refractive index is fn 



Oi-xSi = 6is/ =y i 



to this degree of accuracy, and therefore summing the series 

 of difference equations (as in E. T. Whittaker's tract on 

 the 4 Theory of Optical Instruments ') 



fi K A K ?/l _!%* 02 J 1 1 \ 



-—72 ~9^ ^H'JvKn c i, *'r 





