246 Mr. J. F. W. Herschel on the aberrations of 
(56/2 — 48)^2+ |(56A' — 48) x — 49 ]xVA':=o. 
the independent parts of which being made to vanish sepa 
rately, we find, 
The former of these determines at once the form of the ante- 
rior glass, which must be a double convex or concave of the 
best form, (the curvatures as 6 : 1) in its best position. The 
latter being substituted in X=o gives 
x 3 — i*4x x* — 1 31 25 x x + 1=0. 
all whose roots are real, viz : 
x=— 0*9798, x= + 0-5609, x- +i-8 193 
The first of these values gives the worst possible mode of 
correcting the aberration, the second lens almost exactly neu- 
tralizing the first. The second destroys the aberration by 
the application of a correcting lens whose effect in altering 
the power is the smallest, while the third is that which affords 
the greatest possible power. If we execute the numerical 
computations in the two latter cases we shall find the dimen- 
sions as follows : 
Focal length of the 1st lens 
Radius of its 1st surface 
2d 
Focal length of the 2d lens 
Radius of its 1st surface 
2d 
Focal length of the combination 
These combinations are represented in PL XIX. figs. 4 and 5. 
Whether we ought or not to aim at the rigorous destruc- 
tion of the aberration of rays parallel to the axis, the use 
zd Case. 
3d Case. 
+ IOOOO 
+ 10X00 
+ 5833 
+ 5833 
—35-COO 
— 35 ' 000 
+ 17-829 
+ 5497 
+ 3-688 
+ 2054 
+ 6291 
+ 8-128 
+ 6-4°7 
+ 3‘474 
