242 Mr. Barlow on the rules and principles for 
refraction be as i to i + a ; that is, let the refractive index be 
1 + a ; then it is shown in the work above quoted that the 
aberration, in this medium^ will be 
^ a{d-{-ry d +{a+ 2) r 
(ad — r)* (a + i) d zr 
This expression is for the case of diverging rays on a con¬ 
vex surface of a denser medium; but it will apply to the case 
of a concave surface by merely changing all the signs of r. 
For parallel rays, d must be considered infinite ; and for con¬ 
verging rays, d must be taken negative ; so that this expres¬ 
sion is general in all cases where the rays enter a denser 
medium. 
When the rays pass from a dense to a rare medium, the 
ratio is 1 4“ ^; but this, to be rendered symmetrical, must 
be reduced to i: i — 6, where b = • substituting there¬ 
fore in the above, every where — b for a, we obtain for the 
case of diverging rays on a convex spherical surface. 
_^ + r )' rf + (2 — 6) r y 2 _ 
■* (6rf+r)* (i — b)d zr 
And the expression for converging rays on a concave surface 
is precisely the same, except in the sign of the last factor; 
because both d and r changing from positive to negative, 
leave the expression precisely the same, with the above ex¬ 
ception ; it becomes therefore in this case 
bjd'^r'y d' + {z — b)r' 
^ 77 ' - - (S) 
^ (6d' + "')* {i — b)d' 
merely writing d' and r' for d and r, for the sake of distin¬ 
guishing between the two formulae. 
7, Now, in order to find the aberration of a lens, as caused 
by the refraction at the second surface, which is equivalent 
to the rays falling upon the spherical surface of a rarer 
