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1. If a ray of light, or a pencil of rays traverse a system of coaxial lenses,—the lenses being of any thickness, of any focal lengths, and of any refractive indices whateverthe relation between the positions of the focus of the incident and the focus of the emergent pencil, and a formula for the magnification produced by the system of lenses, could formerly be determined only by an exceedingly cumbrous calculation. It was necessary, moreover, to repeat the process for each different system.

For the sake of simplicity it was often assumed that the lenses were indefinitely thin. The laboriousness of the calculations was thereby considerably reduced; but it is clearly a supposition which it is quite improper to make, except under very special circumstances.

In a paper communicated by Gauss to the Royal Society of Göttingen on the 10th of December, 1840,* it was shown how the solution of the problem could be made to depend upon the determination, for each system and once for all, of four fixed points situated upon the axis of the system. These points having been determined, the complete solution of the problem became a matter of simple algebra or Geometry.

* C. F. Gauss Werke. Band. V. Göttingen, 1840.


It is true that the calculation of the position of the four points is somewhat laborious, but the formulæ obtained are symmetrical, although long, and the formulæ for a system of n+1 lenses can be deduced very easily from that for a system of n lenses.

If therefore a table of formulæ be calculated for 2, 3, 4 ... lenses, which can easily be done, the application to any particular system is a question of Arithmetic and Algebra only.

Gauss' method is applicable to any system of coaxial lenses, whatever be the thicknesses of the lenses, whatever be the refractive indices of the media which occupy the spaces between them, and whether the medium in front of the first lens is the same as that behind the last, or not. The problem however becomes much simpler when these first and last media are the same.

One restriction, however, must be made. It is supposed that the angle which any ray makes with the axis, and also the distance from the axis of the point at which it cuts any refracting surface, are so small that their squares may be neglected. This is equivalent to neglecting aberration.

2. Let us consider a number of spherical surfaces, all of which have their centres of curvature upon a certain straight line. This straight line we may call the axis of the system.

If a system of this kind be intersected by a plane which contains the axis, the section will be such as is represented in fig. 1.

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In this figure the straight line A, A....A,, upon which all the centres of curvature are situated, is the axis of the


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