254? Mr. J. F. W. Herschel on the aberrations oj 
nr l__£ £ H — — 1 L 
ly 1/ £ _£_* L" £ £ 
P" ?" P 3 
In order then that this construction should be applicable to 
any useful purpose, the media must be such as to give mo- 
derate values to L, L', L", which will (generally speaking) be 
insured, provided none of the quantities y, y, y, approach 
very near in magnitude to the corresponding values •£>, y, 
or in other words, provided the media differ considerably in 
the scales of their dispersive powers. 
Developement and application of the equations for correcting the 
spherical aberration. 
18. The reciprocal distance of the focus of any combination 
of lenses or spherical surfaces from the posterior surface, is 
universally resolvable into a series of the form 
M + N/+ 0/+ P /+ &c. 
where y is the semi-aperture, or distance of the point of the 
first surface on which the ray falls, from the axis. In order 
then that this may be rigorously the same for rays incident 
on every point of the surface, this must be independent of y, 
and of course we must have 
N = o, O = o, P = o, &c. 
In ordinary telescopes however, y is sufficiently small to ad- 
mit of our neglecting its fourth and higher powers with per- 
fect impunity. Taking the focal length of the telescope for 
unity, if we allow an inch of aperture to every foot of focal 
length, we shall have y = y, y*= - T &c. So that the 
remaining terms may be safely neglected in our present en- 
