234 Mr. J. F. W. Herschel on the aberrations of 
The integration of this equation, in which rand v are given 
functions of x , must be performed on the hypothesis that 
when j=o,/= ( 1 r — where ^ and r have their 
initial values. Dr. Young has given a solution of a parti- 
cular case of this difficult problem in his paper on the Mecha- 
nism of the Eye, in the Phil. Trans, for 1801, p. 32. 
4. The other case of our equation ( c ) which we proceed 
to examine, is that, where the surfaces are finite in number, 
and placed close together, so as to form a compound lens 
infinitely thin in the middle. In this case we have t—o, and 
the equation becomes simply 
/'= *'+ m'J 
which (putting/ 0 = - D, F ; =”r). &=■ g ives at once 
1 12 
by integration 
/( or /„) = jrta f, + ^ *2 + • • • • K — 0 } 5 M 
'n 
If, after passing out of a vacuum through any number n of 
surfaces, the ray emerge again into a vacuum, we have 
P 8 =i, and 
/= ^ 2 <?z + Y'n Pn D ’ (O 
If in this equation we put for <p 2 , &c. their values in terms 
of r , r , &c., and put 
I 2 
/*, C 1 — m . ) = K> f*, ( 1 — &c - 
we shall obtain 
/=;H*. r .+ *z r * + *.V“ D } S 
Suppose now the radiant point to be infinitely distant, or 
D=o, then will /become the principal reciprocal focal length 
or power of the system ; and calling this F, we get 
