240 Mr. J. F. W. Herschel on the aberrations of 
i> 3 +( — 3) r 2 2 ^,+( 2 ^ 3 — — 4^+3) r 2 r j 2 — (i+ m ) ( i *— m)V l 
The expression of in terms of r and r will be 
simplified, if we recollect that when r = r z the lens will have 
no aberration, being in this case merely an infinitely thin 
spherical lamina, equally thick in all parts, through which all 
rays pass without deviation. A / must consequently vanish 
when r = r , and must therefore have r — r for a factor ; 
so that jwQj- |-Q 2 must be divisible by this without remainder. 
Observing this, we get, on making the reductions, 
A/=f* a (f‘— O ; (”) 
provided we take 
a = 2 m 3 — 2 m -j-i; /3 = 4 m 3 -f 3 rz 2 — 3 m; y = 2 m 3 -f $m x 
a! — m 2 -f- 2 m — 2 j 3 '= w 2 + 3 m 
Now, it has been shown that, L being the power of the lens, 
L = (f*— 3 ) ; 2 ) 
consequently the above value of A / reduces itself to 
9. Suppose now we place any number of lenses close -to- 
gether in vacuo, then we shall have, as in Art. 5, 
a i 
a = 1 
then 
(r.-rj {a + bd + cd-} 
