238 Mr. J. F. W. Herschel on the aberrations of 
The same is true of the small alterations in the values of f 
produced by the aberrations. If we denote by </the change 
in the value off due to the action of the n preceding surfaces 
by S'f that due to the action of the («-|-i)th, and by Af the 
total change arising from their combined action, we shall have 
A /=>/+>? 
Now, 1st. to investigate the partial alteration If in the value 
of f arising from the total alteration Af in that of f, we re- 
sume our equation (c); and differentiating its first member 
according to the characteristic $ and its second according to 
A, vve get (/and/' being the only variables) 
$r, m'Af 
2dly. to discover the partial variation V f of/', arising 
immediately from the action of the (w-j-i)th surface, we have, 
by the equation (6) writing V f for Af, and m', r 1 , D',/ re- 
spectively for m, r, D andy, 
S' f— il . m' (m'—i ) (r'+ D')’ { m' r'+ (l +m') D' J 
but, in this case, neglecting the fourth and higher powers of 
the semi-apertures, it is easy to see that 
y'=y(i- fi) 
and substituting this fory and — —A, for D', the equation 
becomes, 
rf=£ ■ m'(i-m') {r'-f-fr'ty{m'r'-A$r} ; 
so that, uniting the two variations, we find 
A/'- — A/= - / 
= A . m! (1 — in') (r'-f—fr'ty . [m 1 r'— { k ) 
If the surfaces be placed close together, or £=o, this becomes 
A/'— -w'A/= j- • m' {i-m!) (r'—fY { m> ^ — (* + m ')J } (0 
