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[[left margin]] 77) [[/left margin]] [[underline]]A Theorem of the aberration of the Rays of Light refracted through a Lens, on Account of its Spherical Figure, by the Rev.[[superscript]]d[[/superscript]] M.[[superscript]]r[[/superscript]] Nevil Maskelyne, F.R.S.[[/underline]] Philos. Trans. 1761. Vol. 52., No. 4.p. 17. Let the Form of the Lens assumed, in the Investigation of the Theorem, be a Meniscus, the Radius of whose convex surface is greater than that of its concave Surface; and the Center of whose two Surfaces lies on the same side of the Lens, as the radiant Point, from which the Rays diverge, that fall thereon. The Ray falling on the extreme Part of the Lens will, after Refraction, diverge from a Point before the Lense, nearer thereto than the geometrical [[strikethrough]] focus [[/strikethrough]] Focus of Rays diverging from the same radiant Point, and passing indefinitely near the Vertex. Let Q express the Distance of the radiant Point, before the Lens, from its Vertex; R the Radius of the Concavity of the Surface, on which the Rays [[strikethrough]] fall [[/strikethrough]] first fall; and r the Radius of Convexity of the second Surface; F the principal Focus, or the Focus of parallel Rays, which will be on the same side of the Lens, as the incident Rays, because R, the Radius of the Concave Surface, is supposed less than r, the Radius of the convex surface. Let the Ratio of m to n be the same with that of the Sine of Incidence to the Sine of Refraction of Rays passing out of the Air into Glass, and let Y express the Semidiameter of the Aperture of the Lens; the Angular Aberration of the Ray falling on the Extremity of the Lens, and another Ray or Line, suppose to be drawn from the same Extremity of the Lens, to the geometrical Focus of Rays diverging from the same Radiant Point, and passing indefinitely near the Vertex of the Lens, expressed in Measures of the Arc of a Circle to the Radius of Unity, will be [[see transcription note for the equations]] ______________ m^3-2m^2n+2n^3*Y^3 ------------------ _____ (m-n)^2*2m*F^3 + ____________ mn+4n^2-2m^2*Y^3 ---------------- ___ m-n*2m*F^2r + ____ m+2n*Y^3 -------- m*QFr - _____________ 4n^2+3mn-3m^2*Y^3 ----------------- ___ m-n*2m*QF^2 - _____ 2m+2n*Y^3 --------- m*QFr + _____ 3m+2n*Y^3 --------- 2m*Q^2F Where R, the Radius of the first Surface, is exterminated; and r, the Radius of the Second Surface, is retained. Or, exterminating r, the Radius of the second Surface, and retaining R, the Radius of the [[strikethrough]] Second [[/strikethrough]] first Surface, the angular Aberration is likewise expressed by m^2*Y^3 ------------- _________ (2*m-n^2)*F^3 - ____ 2m+n*Y^3 ------------- ___ 2*m-n*F^2R + ____ m+2n*Y^3 --------- 2m*FR^2 + ____ 3m+n*Y^3 --------- ___ 2*m-n*QF^2 - _____ 2m+2n*Y^3 --------- m*QFR + _____ 3m+2n*Y^3 --------- 2m*Q^2F It may be proper to remark, that, as in these Theorems, the principal Focus is supposed to lie before the Glass, as well as the radiant Point, to adapt the Theorem to other [[strikethrough]]uses[[/strikethrough]] Uses, if the Lens be of such a Form, as that its principal Focus lies bhind the Glass, F must be taken negative: Likewise if the Rays fall [[strikethrough]] converging on the Lens, or the Point, to which they converge, lies behind the Glass, Q must be taken negative: Lastly, if the first Surface be convex, R must be taken Negative; and if the second Surface be concave, r must be taken negative; and if, after all these Circumstances are all allowed for, the Value of the Theorem comes out positive, the Aberration is of such a Nature, as to make the Focus of the extreme Rays fall nearer the Lens before it, than the geometrical Focus, or father from the Lens behind it: But if the Value of the Theorem comes out negative, the Aberration is of such a kind as to make the Focus of the extreme Rays fall farther from the lens before it, than the geometrical Focus. (Continued on p. 79.)
Transcription Notes:
For the equations, I have introduced modern parentheses to make these complex equations which a both a sum of six fractions, each which has an algebraic numerator and denominator. I have also rendered squares and cubes as x^2 and x^3 for compactness, and used the modern * for multiplication (where use uses an 'x'). I am also not sure what the convention of putting a bar over an algebraic function means (average?), so have just recorded where they are.
09-14-14-BW I think the overbars are the writer's version of parentheses. I have transcribed the equations as presented with the exception of adding parentheses where the writer uses the overbar with the downstroke because I can't replicate the downstroke aesthetically.
I have preserved the original transcriber's transcription of the equations below, except that I had already added a few parentheses to them in keeping with my thought that the bars are meant to group items as in parentheses. But I got lost in editing them. Sorry. The SI Transcription team can decide which of the two methods to keep, if either.
First set:
(([[overbar]]m^3-2m^2n+2n^3[[/overbar]])*Y^3)/(([[overbar]]m-n[[/overbar]])^2*2m*F^3) +
(([[overbar]]mn+4n^2-2m^2[[/overbar]])*Y^3)/(([[overbar]]m-n[[/overbar]])*2m*F^2r) +
(([[overbar]]m+2n[[/overbar]])*Y^3)/(m*QFr) -
(([[overbar]]4n^2+3mn-3m^2[[/overbar]])*Y^3)/(([[overbar]]m-n[[/overbar]])*2m*QF^2) -
(([[overbar]]2m+2n[[/overbar]])*Y^3)/(m*QFr) +
([[overbar]]3m+2n[[/overbar]])*Y^3)/(2m*Q^2F).
Second set:
(M^2*Y^3)/(([[overbar]]2*m-n^2[[/overbar]])*F^3) -
(([[overbar]]2m+n[[/overbar]])*Y^3)/(2*([[overbar]]m-n[[/overbar]])*F^2R) +
(([[overbar]]m+2n[[/overbar]])*Y^3)/(2m*FR^2) +
([[overbar]]3m+n[[/overbar]])*Y^3/(2*([[overbar]]m-n[[/overbar]])*QF^2) -
([[overbar]]2m+2n)*Y^3/(m*QFR) +
([[overbar]]3m+2n[[/overbar]]*Y^3)/(2m*Q^2F).
