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Extra info for A Note on the Simple Device for Increasing a Photographic Power of Large Telescopes (1920)(en)(4s)

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P0u)(ccy0 + q0M sinc0). 54) and p = A\p0\x0 + Bp0, <\ = ^|Po|yo + Bq0, l = Aa + Bl0. f. 55) Check Formulas p2 + q2 + I2 = n2, xq- yp = x0q0 - yoPo. 6 Numerical Example g. Paraxial Rays 37 The tracing formulas for paraxial rays are obtained from the above exact formulas by neglecting quadratic and higher terms in x, y, p, q, and φ0 relative to unity. This leads to the formulas M Γζ = J° du n(l-p0u)2' z — z, x = (1 - p0z)(ctx0 + Mpo), y = (l- p0z)((xy0 + Mq0), ( 3 · 57 ) P= 1 [(αί " % o | * o + (1 + |po|Afi)p0]9 1 - p0z q =z [(«r - b)\p0\y0 + (1 + i — p0z \p0\Mt)q0l l = n9 where a = 1 + n0p0M, t = £n(l - Poz), n = n(z\ b = εη0, η0 = n(0).

Continue iteration to determine limiting value of u. Retain, for later use, limiting values of M, 1 — p0u, and other parameters. d. 52) a = cos Θ — ß n0 [from Eqs. 30)] and x = (1 — p0u)((xx0 + p0M sincö), y = (le. p0u)(ccy0 + q0M sinc0). 54) and p = A\p0\x0 + Bp0, <\ = ^|Po|yo + Bq0, l = Aa + Bl0. f. 55) Check Formulas p2 + q2 + I2 = n2, xq- yp = x0q0 - yoPo. 6 Numerical Example g. Paraxial Rays 37 The tracing formulas for paraxial rays are obtained from the above exact formulas by neglecting quadratic and higher terms in x, y, p, q, and φ0 relative to unity.

73) From this it follows that r = l(V3-l). 74) Evidently at this point φ = π/2. The value of Θ at F is obtained from Eqs. 73), the result being Thus 2Θ = 150°, showing that the ray, as expected, meets the opposite surface of the lens exactly on the axis. 7 41 Other Examples curved and located on the surface of the lens. However, by a slight generalization of Eq. 59), an index function is obtained having two parameters while still allowing exact evaluation of the integral in Eq. 8). Although the imaging obtained is no longer sharp, the lens designer is provided with two degrees of freedom whereby he can attempt to select a form that meets practical constraints and also gives a reasonable correction of aberrations.

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