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a formula which expresses the magnification in terms of u.

59. We may find a formula for m in terms of v in a similar way, or we may deduce it from Art. 58.

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60. Helmholtz' Formula for the magnification. From Art. 38, we have for the first surface

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(ii),

Using a similar notation for the second surface, we get

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61. Points of unit magnification.

If, in the formulæ for the magnification, we put m=1,

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These values of u and v determine two points on the axis, which we will denote by H and H' respectively. They are conjugate points; and are such, that if an object lie in a plane through either of them perpendicular to the axis, an image of exactly the same size as the object will be produced on a corresponding plane through the other point.

62. These two points Hand H' are of the very greatest importance in the discussion of the path of a ray of light through a thick lens, or through a system of thick lenses. They are fixed points, whose positions depend entirely upon the constants of the lens, and they may therefore be used very conveniently as origins, with reference to which the positions of other points may be reckoned.

Gauss was the first to introduce them into the problem. They were called by him Haupt-puncte or Principal Points. The planes through them, perpendicular to the axis, he called Haupt-ebene or Principal Planes.

We may define these Principal Planes and Principal Points as Planes and Points of Unit Magnification.

63. Formula for the magnification, when the foci are the origins from which distances are measured.

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u and v being measured from F and F'' respectively.

(ii),

1

Hence we see again that uv =

2

(Art. 56). A*

64. Formulæ for the magnification when the Principal Points are the origins from which distances are measured. The principal points are given by

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u and v being measured from H and H' respectively.

65. The formula connecting u and v when these distances are measured from the Principal Points.

We have, from Art. 52, the relation

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u and v being measured from the vertices.

Hence, if we transfer the origins to the principal points,

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therefore (Av+ C− 1) (Au + 1) = A Cu + C−1;

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that is, the reduced distance between either Focus and the corresponding Principal Point is equal to the reduced distance between the other Focus and the other Principal Point; and, moreover, the Foci are either both between or else both outside the Principal Points.

The absolute distances are

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and are therefore

also equal to one another, if the extreme media are the same; but they are not necessarily equal, if the extreme media are different.

67. Definition. The distance H'F' is called the Focal Length of the Lens.

We may however take the reduced distance for the Focal Length, if it be distinctly understood that we do so. The two are the same when μ=1..

therefore C' is a fixed point for all positions of P in the plane PN.

47. If the point N move along the axis to an infinite distance from the lens, the rays which proceed from it, in the limiting position, will before incidence be parallel to the axis, and after emergence will meet at a certain point N' on the axis.

Again, if the point N' move along the axis to an infinite distance from the lens, the emergent rays which converge to N' will, in the limiting position, be parallel to the axis, and must before incidence have issued from an origin of light at a point N situated upon the axis.

The limiting position of N' as N moves off to an infinite distance, and the limiting position of N as N' moves off to an infinite distance, are called the Principal Foci of the lens. They are commonly referred to simply as the Foci, and are denoted by the letters F" and Frespectively.

Hence all rays which before incidence are parallel to the axis will after refraction pass through the point F", and all rays which after emergence are parallel to the axis must have proceeded before incidence from the point F.

48. If we denote by I and I' respectively the infinitely distant points towards which N and N' move, it follows that I and F', and F and I' are pairs of conjugate points.

49. The planes through the foci F and F" perpendicular to the axis are called the Focal Planes.

50. The planes conjugate to the Focal Planes are at an infinite distance.

Hence, if the luminous point be on a Focal Plane, it follows that all the rays which proceed from it will, on emergence, be parallel to one another.

Also, if an image fall on a Focal Plane, it follows in the same way that the incident rays must all have been parallel to one another.

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