ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
85 
A — b ^ Aq -f- b 2 1c 2 4" 1c 3. 
B = 7q -f- b 2 7< 2 “1“ 7j 3 "4“ 1c 1 4* ft 2 ^2* 
C = ft x 7^ + ft 2 7i 2 . 
A' = fcj 2 Aq + fr 2 2 1c 2 + & 3 * 
B' = 7q 2 + 7>/ 2 ^2 + 7/ 3 4- 2 ft x 7q 7q + 2 « 2 & 2 & 2 * 
C' = ft ! 2 Aq + ft 2 2 7; + 2 ft! &! 7*! + 2 ft 2 b., h 2 . ~ 
D = Gq 2 7q 4- ft2 2 h 2- 
5 1 = AQ> + BQ 3 + C. 
5 2 = A' Q3 3 + B'Q 3 2 + C'Q 3 + r>'. 
S 2 is to be equated to zero and the cubic equation for Q 3 solved, as 
the spherical aberration along the axis is nil. By suitable selection of 
the proper kinds of glass S 15 the expression for the satisfaction of the 
sine law can be indefinitely small. Along with the values of Q 3 got by 
equating either S x = 0, or S 2 = 0, there must be taken 
fi = hi + Aq Q 3 - 
f 2 = h 2 4- 1c 2 Q 3 . 
Qi = fti + &! Q‘3' 
Q2 = a 2 4 - 7>2 Q 3 * 
Aud for control, 
S 1 =.SQ«/c 
, 1 n 
= <T 3 + 
/} = ^3 + 7c 3 Q 3 . 
S 2 = 2 Q 2 */«. 
u 3 
ft 3 = + 
0*1 = ft'3 + 
Qr 
n 2 
2 - 1 Q 3 + M ‘- n - 
n., 
2 
n, — 1 
11. y 
n , n., 
q 2 . 
11 , — w,, _ 77 n — ; 
Qs H Q2 H 
72 , Wo 77 n 72 , 
-Qi* 
1 1 
— with regard to — is the distance of the intersection point of the 
ft* ft K 
ray refracted from the (k — l)th to the Kth surface with the axis, and is 
measured from the vertex of the Kth surface. 
If p 15 p 2 , p 3 are the reciprocals of the radii r v r 2 , r 3 , then 
1 Qi , 
p 1 = — = r °q* 
7*i 7? 0 
P 2 = 
7 *o 
q 2 
+ ft 2 * 
_ 1 _ Q3 , 
- t: “ t + ^ 
Controlling equations for h aud 1c arc 
7ij 4- 7i 2 + 7/ 3 = a' 3 — 
Aq + ^2 + ^5 
0. 
