242 
Mr. Barlow on the rules and principles for 
refraction be as 1 to 1 a ; that is, let the refractive index be 
a ; then it is shown in the work above quoted that the 
aberration, in this medium , will be 
_^ g(rf + r) a rf-Ha-f 2 )r y * / % 
F ( a d — r) z (a + i ) d X 2 r ~ " v ''* / 
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 -f- a : 1 ; but this, to be rendered symmetrical, must 
be reduced to 1 : 1 — 6, where b = -~~ a : substituting there- 
fore in the above, every where — b for a, we obtain for the 
case of diverging rays on a convex spherical surface, 
— b{d -f- r) a d -f (2 —b)r y^_ 
F {bd-\-r) z (1 —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 
b(d'+r') z d' + { 2 
A 
b)r' 
( b d‘ + "') 1 
(1 —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 
