114 



in which 



C__2 - ^J!!±i_/«, . _| l-3.5...(2i+ 1) z \ 



*'- WO (2 + ^)2' + * V 2'i-*^2.4.6.,.(2J + 2) 2t + 4/ ■ ^ ^ 



Each of these forms gives, when we neglect ;?'*, the following 

 approximate expression for the characteristic function F of a system 

 of ordinary rays, symmetric about the axis of «, 



l - 2 _ H/(0) 4- '* _ <~= + ^'".y _ i'+^6w,y iz+8w ,)V _ 



l^ ~ ^2(z + A) S{z + AY I6(z + Af "^ &(z + ^)' ' ^ ' 



in which, ;; = a;* + ?/*, and ^0), yj, w,, w^, are constants in the 

 development of the connected function W, such that when we neglect 

 the eighth power of the sine of the angle contained between a near 

 ray, and the axis of revolution of the system, we have 



j= mo) + ^("' + ^') + u>, (^' + fi-^f + w, («^ + fi^r ' (D") 



a, ^, being, as before, the cosines of the angles that the near ray 

 makes with the axes of x and y, to which it is nearly perpendi- 

 cular. 



Verification of the approximate Integral for Systems of Revolution. 



7. The approximate expression (C") for the characteristic function 

 of an optical system of revolution, admits of extensive applications: 

 it is therefore useful to consider other methods, by which it may be 

 obtained or verified. An immediate verification may be derived 

 from the partial differential equation (P') of which (C") ought to be 



