[ *9 ] 
which will be on the fame fide of the lens, as the in- 
cident rays ; becaufe R, the radius of the concave fur- 
face, is fuppofed lefs than r, the radius of the convex 
furface. Let the ratio of m to n be the fame with that 
of the fine of incidence to the fine of refraction of 
rays pafiing out of air into glafs, and let Y exprefs 
the femidiameter of the aperture of the lens ; the 
angular aberration of the ray falling on the extremity 
of the lens, or the angle made between this ray, after 
being refraCted through the extremity of the lens, and 
another ray or line, fuppofed to be drawn from the 
fame extremity of the lens, to the geometrical focus 
of rays diverging from the fame radiant point, and 
pafiing indefinitely near the vertex of the lens, ex- 
prefied in meafures of the arc of a circle to the ra- 
dius unity, will be # 
m 6 
— 2 m z n -f- 2 « 3 X Y 3 
+ 
inn - f- 4 n a — 2 in* X Y s 
m — ^ X 2»i X F 3 
m — n X 2 m X F* r 
m -f- 2 n X Y 3 
4 »* 
-{- 3 in n 3« l X Y 1 
2 m X F r x 
m — n X 2 m X Q_F* 
2 m + 2 n X Y 3 3 m -f- 2 n X Y 3 
+ 
m X Q.F r 
2 ot X Q! F 
Where R, the radius of the firft furface, is exter- 
minated ; and r, the radius of the fecond furface, is 
retained : 
Or, exterminating r, the radius of the fecond fur- 
face, and retaining R, the radius of the firft furface, 
the angular aberration is alfo exprefied by 
« l XY 2 3 * 2 m -f. n X Y 3 
2 x m — » u x F 3 2 X m — k X F S R 
m -f- o. n X Y 3 
2 in X F K4 
_j_ 3 m + n X Y 3 
2 X m — n X QJ 71 
2 711 +_ 2 «X Y 3 
3 m + 2«XY 3 
m X Q_F- R ‘ 2 m X F 
D 2 It 
