228 Mr. J. F. W. Herschel on the aberrations of 
surfaces : in this respect it has the advantage of most, if not 
all, of the constructions hitherto proposed on theoretical 
grounds. 2dly. That in this construction, the curvatures of 
the two exterior surfaces of the compound lens of given 
focal length vary within extremely narrow limits by any 
variation in either the refractive or dispersive powers at all 
likely to occur in practice. This remarkable circumstance 
affords a simple practical rule applicable in all ordinary cases, 
for calculating the curvatures in any proposed state of the 
data, and requiring only the use of theorems with which 
every artist must be familiar ; and at all events, rendering it 
extremely easy to interpolate between calculated values, 
gdly. That the two interior surfaces approach, in all cases, so 
nearly to coincidence, that no considerable practical error can 
arise from neglecting their difference, and figuring them on 
tools of equal radii. Indeed, for a ratio of the dispersive 
powers a little above the average, they are rigorously coinci- 
dent, and this construction coincides with that of Clairaut 
above-mentioned ; and so nearly is this approach to equality 
of curvature sustained throughout the whole extent of the 
function, that even when the ratio of the dispersive powers 
is so low as 0.75 : 1 (a case almost useless to consider) the 
difference amounts to less than a 40th part of the curvature 
of each. 
§ I. General formulce for the focal distances and aberrations of 
any combmation of spherical surfaces. 
1. Expression for the focal distance of a single spherical surface. 
Let C be the centre of a surface AM (PI. xix. fig. 1), on which 
a ray QM proceeding from a point Q in the axis, is incident, 
