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By Hamilton W.R.

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Dx + = 0; (h − ρ1 ) dx (h − ρ2 ) dy which, when we put dy = dx . tan. ψ, becomes h= ρ1 . ρ2 , ρ2 . 2 ψ + ρ1 . 2 ψ (D ) a formula that evidently coincides with the one that we found before, for the height of the osculating focus. IX. On thin and undevelopable pencils. ] Having examined some of the most important properties of the developable pencils of a reflected system, we propose in this section to make some remarks upon pencils not developable; and we shall begin by considering thin pencils, that is, pencils composed of rays that are very near to a given ray; because in all the most useful applications of optical theory, it is not an entire reflected or refracted system that is employed, but only a small parcel of the rays belonging to that system.

1 + p − (α + γp) }, 2 2 . {1 + q 2 − (β + γq)2 },                 . {pq − (α + γp)(β + γq)},  42 (W ) which contain the solution of the problem. To shew the geometrical meaning of these equations (W ), let us take the vertex for origin, the normal at that point for the axis of (z), and the tangents to the lines of curvature for the axes of (x) and (y); we shall then have p = 0, q = 0, s = 0, r= 1 , R t= 1 , R γ = γ = cos. I, I being the angle of incidence, and R, R , being the two radii of curvature of the mirror; and the formulæ (W ) become h .

I = R(1 − α2 ) = R (1 − β 2 ), αβ = 0, (X ) (h) being the harmonic mean between the conjugate focal distances, so that 2 1 1 = + . h ρ ρ The equation αβ = 0, shews that the plane of incidence must coincide either with the plane of the greatest or the least osculating circle to the mirror; and if we put β = 0, that is, if we choose the plane of incidence for the plane of (x, z) we shall have R = R(1 − α2 ), R > R , so that it is with the plane of the greatest osulating circle that the plane of incidence coincides.

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Theory of Systems of Rays by Hamilton W.R.

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