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664
W. S. VAUGHN AND F. TUTTUS
[j. S. M. P. E.
the solid-line portion of this circle being the object gate proper. Let a be the distance of the center of the circle from the lens L. Let the focal length of L be /. Let C' be the curve representing the image of C formed by lens, L. Let u0 and v0 be conjugate distances of the object and image gates, measured along the optical axis. Only the absolute values of these quantities will be considered.
By the methods of simple geometry, and the use of the lens formula, we obtain the equation of the image curve, C':
(f-a
But since (Fig. 1)
a = u0 — r
. [/ (u0 2r)]«[/ ~ /4r2
which we may write
a r)
,
~ (UP 2r)](f *.]
(3)
The coefficient of x2 is always positive, for real values of u0, r, and /, but the coefficient of y2 may be positive or negative, depending on the relative magnitudes of u0, r, and/. Five cases arise: Two cases for virtual images:
(a) If / > u0 and / > u0 — 2r, then B > o and the image curve is an ellipse. Geometrically, this means that the object circle lies entirely to the right of the /-line, or the plane of the principal focus.
(b) If / = u0 and / > u0 — 2r, then the image curve is a parabola. Geometrically, this means that the object circle lies to the right of the /-line, and is tangent to it.
Three cases for real images:
(c) If / < u0 but / > u0 — 2r, then B < o and the image curve is a hyperbola (one branch is for a virtual image). Geometrically, this means that the object circle is intersected by the /-line.
(d) Iff<u0 and/ = u0 — 2r, the image curve is a parabola (real). Geometrically, this means that the object circle lies to the left of the /-line, and is tangent to it. (See Fig. 1.)
(e) If / < u0 and / < u0 — 2r, then B > o, and the image curve is an ellipse (real). Geometrically, this means that the object circle lies wholly to the left of the /-line.
It is difficult mechanically to shape a gate into a parabola, hyper