ago Mr. J. F. W. Herschel on the aberrations of 
and finally 
i 
— m)r — mer 
i m(i — m){i + e)\m-\- e + me) 2 
r z ( i — m — me ) * 1 * 
Let f denote the reciprocal focal distance for central rays, 
and f + A f the same reciprocal distance for the ray inci- 
dent at M ; then (the aperture being regarded as small in 
comparison with the focal length), the aberration will be 
represented by — and if we put for e its value in the 
foregoing equation, we shall have, 
f = (1 — m) r — m D; 
A/=w(i — m)(r-\- D) 5 jm/*-|-(i-J-m)D J 
(*) 
(b) 
2. Theory of the foci of spherical surf aces for central rays. 
Before proceeding to investigate the more complicated 
cases of the aberrations of several surfaces, we will deduce 
from the first of these expressions the general equations 
which determine the place of the focus of central rays after 
refraction at any number of spherical surfaces ; equations we 
shall have occasion to use hereafter. Let r , r , r , &c. be the 
I 2 3 
curvatures of any number of surfaces A i} A , A , See. which 
form the common boundaries of the media o, 1,2,3, &c. 
(fig. 2) and let the relative index of refraction out of the 
medium o into 1 be — , out of 1 into 2, — l — , and so on ; also 
m m 
1 2 
let p o , y, , y 2 , Sec. be the absolute refractive densities or in- 
dices of refraction out of a vacuum into these several media, 
then will 
//. m ’ u. ’ (j. 
^ O I O 12 0 
Moreover, let t z , 
1 
m m m 
1 2 3 
&c. be the respective 
and so on. 
