The optical theory referred to in the title of this communication is now fully half a century old; and has, moreover, been well expounded in the standard English treatises of Pendlebury and Heath. Still, notwithstanding its elegance and simplicity, and its great practical importance as giving the first approximation to the theory of the great majority of the optical instruments in ordinary use, its filtration into the strata of popular knowledge has been remarkably slow. It seems, therefore, to be worth while to offer a brief summary of its leading principles, freed as much as possible from the detailed calculations which become necessary when the constants of the optical system have to be deduced from the data of construction, and to indicate methods for experimental verification. In giving this summary, I shall omit the demonstrations of some of the propositions, which can be found by those who desire them in the well known Treatise on Geometrical Optics, by Heath

When the curvature of a plane curve continuously increases or diminishes (as is the case with a logarithmic spiral, for instance) no two of its circles of curvature can intersect one another.

On p. 403 of Greenhill's Calculus (2nd Ed.) the following sentence occurs:— “By differentiation of the integral

with respect to A, B, or C we can deduce the results of

For the evaluation of tlie typical form in which f(x) is a linear function, especially when A, B, etc., are given numbers, the method of differentiation does not seem very suitable; be that as it may, it may perhaps be of some interest to investigate a formula of reduction analogous to those in use for the integrals in which Ax2 + 2Bx + C is replaced by a linear function and f(x) is a constant.

If the tangent at a point P on the parabolic curve cy=xn meet the axis of x at M, it is a well-known property that the area between the radius vector OP and the are OP is n times that between the arc OP and the two tangents OM, MP, O being the origin and n > 1. The converse is also true; for taking any point O on a curve as origin and the tangent at O as axis of x, let us seek for the locus of P if the area between OP and the arc OP be n times the area between the arc OP and the tangents OM, MP.

Taking the three addition theorems and clearing away the fractions,

Let

Substitute in (1) (2) (3) the expansions for sinx, siny, cosx, cosy, , dn x, dn y, , and then pick out the coefficients of y in (1), (2), (3). Then equate the like powers of x in the resulting series.

Definition. The isogonals of the medians of a triangle are called the symmedians

If the internal medians be taken, their isogonals are called the internal symmedians or the insymmedians; if the external medians be taken, their isogonals are called the external symmedians, or the exsymmedians

The Series

if convergent, is a particular solution of the Differential Equation

in which

and II denotes Gauss's II Function, D stands for .

What do we mean when we assert that one plane figure is equal to another? In trying to find a satisfactory answer to this question I was led to consider the subjects treated in this paper.

I.—Take the well-known identity

Now if we can transform 4wz into a square we shall have two square numbers whose sum is a square. This will be effected by taking w = p2, z = q2, for then 4wz = 4p2q2 = (2pq)2 and we have

See Mathematical Magazine, Vol. II., No. 5, p.

Take the point P in the arc AB, and let ∠ ABP = θ, then the trilinear equation of the Wallace line of P (p say) is

(a) The lines AP, BQ, CR drawn parallel to the sides BC, CA, AB of the triangle ABC.

In this case p, q, r, by [§ 17] are readily, seen to be concurrent.

By assuming that multiplication by a line is the true operation corresponding to the passing from space of n dimensions to space of n+1 dimensions we may arrive very simply at certain well-known results in geometry of higher dimensions.

From the figure we have at once

similarly

It is known that if from a given point perpendiculars be let fall on the four Simson lines formed from the four triangles made by taking every, three of four points concyclic with the first, the feet of these perpendiculars lie on a straight line proposed to be called the Simson line of the quadrangle formed by the four points; it is also known that this process can be extended. I propose to examine various results connected with these lines.