Hence, in general, the principal foci move in contrary directions, and meet both at the centre and circumference. In the formula just given one of the conjugate foci lies between the principal focus and the surface of the speculum; while in the first set, one lay between that point (r) and the centre. A single formula, however, as the reader may verify, will include all cases, whether of concave or convex specula, or divergent or convergent pencils, provided we admit the use of negative quantities, and interpret the negative sign to mean that the distance affected by it must be measured in a direction contrary to that in the figure on which the formula was framed.
With respect to images, if A 0 be the object and g the focus con jugate to a, then ay will be the image of A a ; and conversely, if a g be the object, A a will be the image, and their proportion maybe easily calculated, for A c : ag: A : c a, that is, asp : p', or as A : which we have seen is the same ratio.
Example 4.—In the concave speculum of two-foot radius, an object is placed within 6 inches of its interior surface : how far will the image appear at the back of the speculum, and how much will it seem enlarged t 1 1 1. 2 Here r=2, in feet, .f= .4-, and since — 3 = we have 1 2 — —= therefore a = 1, or the image will appear a foot behind the A convex side and will be enlarged in linear dimensions as A to a', that is, as 2 to 1 ; in surface 4 : 1; in volume 8 to I.
Example 5.—An object is placed 10 feet distance from a convex speculum of 3 feet radius ; find the position and magnitude of its image.
1 1 2 1 23Here r= 3, therefore 3, whence To, therefore = or 1 foot 3 inches, 8 parts nearly, at which distance in 23 the concavity of the speculum the image will seem to be, and (in linear 30 dimensions) Object : Image : : A : : : 10 : that is as 23 : 3; the surfaces as 529 : 9, kc. Thus the reader with only a moderate know ledge of simple equations will be able to solve all questions relative to the images of objects formed by spherical specula, concave or convex. The images in the last two examples are erect. Generally the image will be erect or inverted according as one of the conjugate foci is between the principal focus and surface, or between that point and the centre ; and this will include all cases, for it is easily seen that in every case one of the foci is in some part of the radius between the centre and surface.
In the preceding calculations, we have confined ourselves to such rays as fall nearly perpendicularly on the reflecting surfaces. The rays which are at a considerable distance from the axis of a spherical speculum are not reflected accurately to the same point as those incident near the axis ; hence arises a diffusion of the reflected rays arising from the sphericity of the speculum and denominated the spherical aberration ; and when measured along the axis, it is called the longitudinal aberration ; but when perpendicular to it, through the focus, the lateral aberration. It will be sufficient in this article to
calculate the amount of these aberrations in the most usual case when the incident rays are parallel, as those which proceed from the heavenly bodies.
Let s to represent a ray falling parallel to the axis en; nn being the intermediate arc of the section of the speculum, D a the reflected ray; if this figure revolve round c a, it is evident that all rays incident on the annulus through which D moves will likewise be reflected to a, which is therefore strictly the focus of that annulus, Now F, the middle point of CB, is the point to which rays falling near the axis are reflected; hence a v Ls the longitudinal and F b the lateral aberration corresponding to the above annulus. To calculate the amount of these we may observe that the angle a n C (of inci dence) is equal to C D a (of reflection), and also to D C a (by the theory of parallels); and since the angles an c, a on, are thus equal, therefore ca = a D. Let Dr be a tangent at D, then a n x and a T D, being respectively the complements of a D c and a C D, are also equal, whence a T= a D, but also c a= a D, therefore a LI the middle point of c T ; and since r is the middle of c n, it follows that a r is the half of B T ; thus the longitudinal aberration is known ; and since the angle r a b is the double of D C R, the lateral aberration is from thence known. Let the angle D C B= 0, and radius c B=r, then c T=r sec. 0 and B T = r (sec. 0-1), hence we obtain the exact values of the two aberrations, namely, the longitudinal = (sec. 0-1), and the lateral r tan. 2 0 rb— 2 (see. 0—I). Ilence in order that the aberrations may be inconsiderable, we ought to have the extreme magnitude of 0, namely, the angle B c d (in fig. 3), also small. On this supposition formulas sufficiently approximate may be deduced from the above and 0 better adapted for practice. For sec. 0 put 1 , and for tan. 2 0 put which are respectively the approximate values ; thence we get, longitudinal aberration= and lateral aberration B = — 2 = 2 of which are evidently very small, particularly the latter. The least circle of aberration is the smallest that would be formed on a card placed perpendicular to the axis near the focus F to receive the reflected rays; now if the intersection g of a reflected ray ny with the final one dg be taken the most remote possible from the axis c B,, it is evident that all the other reflected rays will pass between g and the axis, and hence the perpendicular distance from g to the axis is the radius of the circle of least aberration or diffusion. The question is thus reduced to one of maxima and minima, and may be easily solved in the usual manner by means of the Differential Calculus.