insensible. In this instance, then, it would be proper to attend to the first condition. But in all cases of micrometer measurement, in which the power is not very great, distortion is of little consequence, the object and the wires being equally distorted; but it is important that the object and the wires should be seen distinctly in all parts, without pushing or pulling the eye-piece. Here then the second condition should be insisted on, and the third, if possible, combined with it. For a telescope, which is used for examining the disc of a planet, or any similar observation without instrumental measurement, the third condition is quite as important as the second: the first less so. For common perspective glasses, opera glasses, &c. regard ought to be paid to all three, but especially to the first. I shall take occasion to apply the formulæ to the construction of the camera obscura: the second and third conditions ought there to be enforced, except the sacrifice of the first is very great.

The same causes which divide the effects of spherical aberration into three distinct parts, oblige us to use, for the investigation of these parts, three distinct operations. I shall, therefore, investigate, 1st, the course of the axis of a pencil; 2d, the point of convergence of rays in the plane passing through the axis of the pencil and the axis of the lens; 3d, the point of convergence of rays in the plane perpendicular to the former, and passing through the axis of the pencil.

It is evident that an object will be seen without distortion if its image, exactly similar to the object, be formed on a plane; and then the trigonometrical tangent of the angle, made with the axis of the lens by the axis of the pencil after refraction, will bear to the tangent of the angle before refraction a constant ratio: if the ratio be not constant, its difference from a constant ratio will indicate the degree of distortion. This suggests the following Propositions.

PROP. I. To find the proportion of the tangents of the angles made by the axis of a pencil with the axis of a lens before and after refraction.

Let AF, FG, GE, Fig. 1, be the course of the ray, and let FG produced meet the axis in D: draw FH, GK, perpendiculars to the axis: then our object is to compare the tangents of the angles FAH, GEK; or GK AH to find the value of Let FH = a, BC=t, AB=b, CE=c, FH EK BD= x; and let B, C, X, be the values of the latter quantities when a = 0. Draw FL perpendicular to KG produced. Then

Now I will remark, that we may at once omit t in this expression, though it is larger than the next term which is to be preserved. The reason is

that

is independent of the aperture, and as it is our object to examine,

among the small parts, only those which depend on the aperture, there is no necessity for preserving a small constant term. The product of t and small variable quantities being extremely small, is, of course, to be neglected. It appears, then, that though in the beginning of this and other investigations, we shall be obliged to take into account, we may always expunge it in a subsequent stage. When multiplied by a constant, the product is to be rejected because small and independent of the aperture: when multiplied by a variable, which is necessarily small, the product will be of an order smaller than the quantities taken into account. Hence,

Let the radius of the first surface be r, that of the second s. Then

Now, by the common approximate formula for the principal foci at refraction by spherical surfaces,

This expression may be put into a more convenient form thus. Let

the principal focal length of the lens = F; letf. Then, neglecting

The quantity c-C may be thus calculated. By the known formulæ

for aberration at refraction by one surface (Cambridge Trans. Vol. II. p. 110),

(since the part independent of a must be the value of when a=0, that is,

must = c); and taking the reciprocal, and subtracting C,

Substituting this value of c-C in the expression for the proportion

This, for abbreviation, we shall call {1+Q}. The first term of Q may

sometimes be more conveniently calculated by observing that

PROP. II. To find the proportion of the tangents of the angles after refraction through several lenses.

The proportion of the tangent of the first angle to the second being

B'
C

{1 + Q}, that of the second to the third {1+Q'}, &c., the proportion

necessary to show how Q, Q', &c. are to be calculated. First, it will easily