16 PROCEEDINGS OF THE CANADIAN INSTITUTE. 



determines whether the image is erect or inverted, the sign of 



U)' 



— being positive in the former case, and negative in the latter. 

 •w 



27.* The method may also be applied to determine the spherical 

 aberration of mirrors. 



Thus in the case of a concave mirror if distances are measured 

 from the centre O, and if the incident ray PI is reflected at I so as 

 .to cut the axis after reflection in P", we know that 

 _/sec a /sec o. 



P P 



where a is the angle AOI, OP = p, and OP" = p". 

 But, P' being conjugate to P, we have 



P V 

 where OP' = p' . 



Hence, if the separated axes cross at O, as in Fig. 15, whilst PP' 

 ■always passes through X ( — f-,f), PP" always passes through Y 

 ,( — f sec a, /sec a). P'P" on the y axis will accordingly represent 

 the longitudinal aberration, whose direction is seen from an inspec- 

 tion of the figure to be from O to A except when P lies between F 

 and G. 



The value of the aberration may be determined by comparing the 

 similar triangles POP', PFX, POP", PGY. Thus . 



P'P" _p" — p' __ /sec a /^ 



P P p-\-f sec a p -\- f 



ijt( 1 cos (X) 



whence we get P'P" = ^-^ — - , the ordinary expres- 



(P +/) (poosa-\-f) 



sion. 



If PF = d, and PG = /\, we also have 



p" — p' /\ — p d — p 

 ■ ~~V A ^ 



/{ 1 \ 

 Therefore p" — p' =z n^ I ) . 



It may be remarked that FG is the principal longitudinal aberra- 



* This section was added December, 1884, 



