16 PROCEEDINGS OF THE CANADIAN INSTITUTE. 
‘determines whether the image is erect or inverted, the sign of 
Uy 
I ee pS eeatee noes 
—being positive in the former case, and negative in the latter. 
@ 
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 
_ fseca | fseca 
| p p 
where a is the angle AOI, OP = p, and OP” = p”. 
But, P’ being conjugate to P, we have 
ee ea 
a i ae 
where OP’ = p’. 
Hence, if the separated axes cross at O, as in Fig. 15, whilst PP’ 
always passes through X (—/,/), PP” always passes through Y 
(—fseca, fseca). 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 
PP” p'—p _—s fseca ‘i 
p p pfseca ptf 
; pf (1 — cos 4) 
whence we get P’P’ — - : : 
< (p +f) (p cosa +f) 
, the ordinary expres- 
sion. 
If PF = d, and PG = J, we also have 
Pee PAS wee =P 
Pp AN d 
l l 
Therefore p” — p’ = p*(—~— - ). 
A 
It may be remarked that FG is the principal longitudinal aberra- 
* This section was added December, 1884. 
