243 
determining the dispersive ratio of glass, &c. 
medium, the latter expression must be employed, in which 
therefore d' is not in efinite as in the former case, but is 
dependent upon the refraction at the first surface of the 
lens, being the distance to which the rays converge in con- 
sequence of that refraction ; d' therefore in this case, by a 
well known expression for the focus of the rays at one 
surface, is # __ ( a + 0 dr _ _ _ / 6 \ 
a d — r \ / 
Where d must be taken positive or negative accordingly 
as the rays first diverge or converge, and r must be positive 
or negative as the first surface is convex or concave, and this 
value of d' substituted in equation (5), will give that part of 
the aberration which depends upon the rays, impinging on 
the second surface. But there is also another part depending 
upon the aberration of the first surface ; for as the rays in 
consequence of the first aberration do not all converge to the 
distance d ' , whereas we have computed the second case as if 
they did, there will be an aberration on that account. 
Let the aberration produced at the first surface be x ; then 
the consequent aberration at the second surface will be 
(Wood’s Optics, Art. 405.) 
(1 — b)r' 
(b d' -{■ r 1 ) 
7 X X 
And hence the entire aberration produced with diverging rays 
b y a convex lens from a distance d, the radii being r, r ‘ , will 
be expressed by 
a (d + r) a d -f (a + 2) r (1 -~b) r ,z J_ 
(ad — r) a (a 1) d (b d' -f r ') 1 2 r 
, Hd'+r 1 )* d'+(2 — b)r' i_ 
“• (bd‘+r l ) z ) (1 — b)d‘ ir' 
And by substituting for b its value and making — = c t 
= q, this reduces first to 
i-r — d, and - 
r ’ r 
