260 Mr. J. F. W. Herschel on the aberrations of 
Table 2. Coefficients of equation (A) for correcting the 
aberration of diverging rays, and //.' as in Table 1. 
ZJ — O'^O 
°'55 
o'6o 
6 6247 x r 
6 6247 
6-6247 
— 17 6627 
— 19-8254 
— 22 6523 
— 3-2618 x r'— 0 
— 37880 
— 39 * 4 2 
0*65 
070 
6 6247 
66247 
— 26-4274 
—31 6243 
— 4-2404 
— 4-5666 
075 
6 6247 
— 39'°979 
— 4 8928 
23. If these equations be combined, we shall obtain the di- 
mensions of an object-glass free from aberration, both for 
celestial and terrestrial objects, provided we restrict our 
views to objects situated in the prolongation of the axis of 
the telescope. The arithmetical operations necessary for 
determining the values of r and r 1 being executed, we shall 
find that the resulting quadratics admit real roots, and that in 
consequence there are two sets of curvatures assignable to 
the surfaces, which satisfy the algebraic conditions. The 
values of the first set are however objectionable, as they will 
be found to correspond to meniscus and concavo-convex forms 
of the crown and flint lenses respectively, of great curvature, 
and placed together as in PI. XIX. fig. 7. Those of the other 
correspond to moderate curvatures and very convenient forms, 
as represented in PI. XIX. fig. 8. The values of r, i J , in this 
series, and those of r 2 and r • the curvatures of the posterior 
surfaces, deduced from them as well as those of L, L', the 
powers of the lenses are set down in the subjoined table, in 
which (for simplicity) we have taken / = 1. 
