243 
compound lenses and object-glasses. 
The aberration of such a lens being computed (for a given 
power L) will be found to equal — ^y 2 L. Let this be called 
a and we shall have the proportional aberrations of the fol- 
lowing lenses as below : 
Plano-convex or concave, plane side first . 4*2 x u 
Do. curved surface first . . 1*081x0 
Double-equi-concave or convex . . . 1*567 xw 
The aberration, for parallel rays, of a double lens, of which 
the first glass has a and b for the curvatures of its surfaces, 
and the second (of the same substance) a' and b', is repre- 
sented by 
y\ [L (7 — 6ab-\-Qjb 2 ) ) 
2 4 (L+L') ;i | -pL'(7tf f2 — 6a'b'-\-2jb ' 2 — J 
In this if we suppose a = h . 2 L, a'= hi. 2 L', which give 
b = (/z — 1) . 2 L, b 1 — ( h ' — 1 ) . 2 L' 
and suppose moreover, x = and 
X = ( 28 h ! 2 — 48 h ’- f 27 ) x 3 + ( 33 — 49 h ! ) x * 
+ 13 x . -j- (28 h " — 48 h -p 27), 
we shall have, for the expression of the aberration of the 
compound lens, 
y'( L+L') x _ 
6 •(!+,)* — “• 
The aberration of the best single lens of equal power is, as 
we have already found, — 71 -y* (L+ L'), and comparing the 
two, we have 
ft 7 x 
a , “ 45 * ( I+*) 3 ’ ( P ) 
11. If we would destroy the aberration, we have only to 
put X = 0. As this cubic equation must have at least one 
mdcccxxi. I i 
