244 ATr. J. F. W. Herschel on the aberrations of 
real root, it follows that whatever be the proportion of the 
curvatures of the surfaces of two thin lenses placed close 
together, it is always possible to adjust their focal lengths so 
as to produce a combination free from spherical aberration ; 
and the same is true if the lenses be formed of different 
materials. 
To take an example or two; suppose the first and third 
surfaces on which the light falls to be plane, and we have 
a — a'= o, and consequently li = h'= o, So that the equation 
X = o becomes 
27 * 3 + 3 + 2 7 — ° 
whose only real root is x — — 1'392. Hence, if we take 
L= + l, we have L-{- L/= + 0-392. So that the power of 
the compound lens is about ^ that of the first glass and of 
an opposite nature, which, though moderate, may not be too 
low to be of some use. 
If an object be placed in the focus of parallel rays so formed, 
the rays it sends to every part of the surface will emerge 
rigorously parallel. Such an object will therefore be seen 
by an eye on the other side with as much distinctness as if it 
were a real object at an infinite distance, subtending the same 
angle, and the combination may thus be used as an eye- 
glass or magnifier, as well as an object-glass, only reversing 
its position with respect to the eye. 
12. Let a — b'= 0, h = 0, li' = 1, or suppose the first and 
last surfaces plane, and our equation X =0 becomes 
7 x 3 — i6i 2 -}-i3 x -j- 27 = 0 
which has but one real root, oc = — 0 85 17, or 
L'= — L. 0*851 7; L-f- L'== L x o’1483 
The lenses then must be of opposite characters, but the 
