232 Mr. J. F. W. Herschel on the aberrations of 
and the equation /'= <p' — m! D' becomes 
f'=<p'+ TZfl- (0 
This is in fact an equation of differences between the con- 
secutive values of/, and the general value may therefore be 
obtained by integration, or its particular ones deduced in suc- 
cession from each other, when the integration is impractica- 
ble from the values assigned to m, <p t and t. The greatest 
simplification it appears to admit, is its reduction to an equa- 
tion of the 2d order and first degree, which may be performed 
by assuming 
j u ' t 
when the equation will become, after the necessary reductions, 
u 
1/ /-/ 1 \ / \ 
o=u' f- T )« +K 
It will not be necessary to examine particularly all the inte- 
grate cases, or to discuss at present the form of the general 
value of u or/ in terms of <p, m , and t. This latter subject 
is elegantly treated by Lagrange, in a Memoir “ Sur la 
Theorie des Lunettes,” in the collection of the Academy of 
Berlin, (Acad. Berl. 1778), to which we may refer. We 
need only remark that, whatever be its integral, it must 
necessarily be of the form 
M— ND 
O— PD 
The original distance of the first radiant entering as the arbi- 
trary constant, and being therefore always involved in the 
same simple algebraic form, whatever be the number and 
position of the surfaces. 
3. Two cases of the equation, however, are worthy of a 
more particular examination. The first is, when the number of 
surfaces is infinite, and the intervals separating them infinitely 
