312 
Transactions of the Society. 
Having found/ the required focus, the radius is easily deduced by the 
formula r' = f (/x — 1). 
In assigning a value to y we must know the diameter of the lens. 
It may be found thus : — having drawn the meniscus and found the 
point B or F and having laid off the angle of aperture at C (fig. 32) 
through the point where the extreme ray from C meets the curve A, 
draw BE. E being the limiting ray it determines the diameter of 
the lens and consequently the value of y. 
If it is required to further diminish the aberration it can be accom- 
plished by crossing the second lens. As stated above, it is not neces- 
sary to do this when the lens is of 1'62 ref. index. A piano flint 
is however a good deal better than a crossed crown, the difference of 
the coefficient in favour of the flint being * 22. Therefore if it was a 
matter of equal cost between a crossed crown and a piano flint, the 
flint should be chosen, and as the meniscus is aplanatic, whether made 
of crown or flint, it might be cheaper to make the meniscus of crown 
and combine it with a piano flint. The combination would yield 
a result as far as aberration was concerned almost equal to that of a 
doublet composed wholly of flint. As crown glass has a green tint, 
where the colour of the light is of importance flint glass only should 
be used. 
To find the radii r and s of a crossed lens of given focus, it is 
only necessary to multiply the focus by the constants H and K 
thus : — 
r = H/ 
• = K /. 
For glass of ref. index 1*516; H = *5935, and K = — 3*944. 
For glass of ref. index 1*62; H = * 653, and K = — 12-06. 
In a crossed lens the flatter curve always faces the meniscus. 
It should be remarked that the formula for the nodal point given 
above, n = -, is not strictly accurate for a crossed lens, but it is 
abundantly so for my purpose. The distance n is measured from 
the flatter curve into the substance of the lens, similar to the 
piano. 
The spherical aberration may be considerably further reduced by 
placing a second aplanatic meniscus next the first, so making the 
condenser a triple (tig. 35). 
The formulae for computing the radii of the second meniscus are 
precisely similar to those of the first. 
Let Q and Q r be the terms of the foci of the second meniscus 
corresponding to P and P' of the first. 
As far as the second meniscus is concerned we have merely to 
regard the light as emanating from B and neglect altogether the 
presence of the first meniscus. 
