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NOTE ON ASPHERICAL LENS SYSTEMS
BY WILLARD J. FISHER
In the paper by L. Silberstein, on aspherical lens systems, J.O.S.A.,11, p. 479-494; 1925, the statement is made, p. 483, that to his knowl-edge the only previous method of closing a simple lens, whose frontsurface is generated by revolving a Cartesian oval, is by means of aspherical concave surface centered at the focal point of the ovaloid.In Eq. (7), p. 486, this case corresponds to makingfi=f 2 .
But if lifi =f2, the second surface is again a hollow sphere, with oneof its aplanatic points at F1, the other at F2 , (Fig. 5, though withoutlimitation to object at infinity).
This combination of surfaces is not aplanatic, by Eq. (15) and theparagraph following. The aplanatic condition being /sin = const.,the deviation from aplanatism for the front surface is proportional tothe departure from constancy of 1/(1 -cos u/ju), in whichu1 = 01F1P1.If the back surface is the aplanatic spherical surface mentioned, andU2 is the angle at F2 , then /.t sin u 2 =sin .
The aplanatic condition can equally well be written sin u/l = const.Then the departure from constancy is proportional to cos u/A=_Vtu. 2 - sin2u2.
For quartz glass and G light, 4340, ,u= 1.467. If cos u changes fromcos u=1, =00 , to cos u=O.99, =8'06', then 1-cos u,u changesfrom 0.320 to 0.327, or 2.2 per cent. Simultaneously u2 changes from0° to 530'. 2 tan 530'=f/5.2, roughly.
Such a quartz lens, computed for the transmission band of a silverfilm, might be of use in eclipse photography.
It may be mentioned that the writer has not seen any mention of thefact that a hollow sphere has aplanatic points. The above is a demon-stration of the theorem, which was however discovered in an attemptto invert the Weierstrass construction, some years ago. As the theoremfor a solid sphere is of great importance in microscopy, its inverse mayequally become of technical importance.
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