2 57 
compound lenses and object-glasses. 
another condition quite independent of the correction of the 
spherical aberration, and which at first sight seems likely 
greatly to increase the difficulty of the investigation, should 
on the contrary tend so remarkably to simplify it. 
In general, when the focal lengths are assumed, there will 
be as many unknown quantities r, i\ &c. as there are lenses, 
and the aberration for parallel rays may therefore be de- 
stroyed in a great variety of ways, some more, some less 
advantageous. If, for example, we limit the figure of one of 
the lenses in any way ^as if we assume it plano-convex or 
concave,) or assign equal curvatures to both its surfaces, &c. 
such limitation is equivalent to assigning given values to 
both its radii, and the terms depending on that lens in equa- 
tion ( w ) pass into the given part of the equation. 
20. Let us next consider the equations (w) and (z;). The 
latter does not involve the radii of the lenses, but only their 
powers, being in fact when developed. 
o= (2m +3) L -f- (2m / +3) L'-{- (2m"-{-3) L"+ &c. ; (x) 
In a double object-glass, this equation will be incompatible 
with the equation o=p L-j-/>' L' expressing the condition of 
achromaticity, and must of course be sacrificed, the latter 
being of paramount importance.* It is in fact a very secon- 
dary consideration to satisfy this condition in telescopes. In 
the microscope, however, where D is necessarily a quantity 
* Unless such a peculiar adjustment of the media should take place as to reader 
the two conditions identical, which would give 
P P_ Qr P u - _ P ft 
2m + 3 — 2 3^ + 2 3f*'+2* 
It is a mere matter of curiosity to look for media satisfying this equation. Fluor 
spar combined with rock crystal comes very near it, but among bodies adapted for 
object-glasses there are probably none to be found except such as would result from 
mixtures of different liquids. 
