The diagrammatic figure illustrates the principle involved. The vertical full lines represent ordinates of a curve of which the

abscissae are Oa1, Oa2, etc. Let now all these vertical lines be rotated round the points O, a1, a2, etc., while retaining their original lengths, through the same angle θ. The curve passing through the new positions of the end points is now to be referred to the tilted axis Oy0 and the line OAB, drawn at right angles thereto. If we put OC = X, CD = Y, and let X′, Y′, be the corresponding coordinates with reference to the tilted axis we get

If a circle has double contact with a parabola, the points of contact being P and P′, its centre, G, is the point of intersection of the axis with the normals at P and P′, and the perpendicular from G to PP′ is the subnormal GN, which is half the latus rectum.

The property of a conic: “if the projections of K, any point on a tangent, on the directrix and the focal radius of the point of contact be I and U respectively, SU = e. KI,” is known as Adams's Property. The statement of the above property in terms of the polar coordinates of K is the equation of the tangent. Hence, if the eccentric angle of the point of contact is a

or = cos (θ — α) + e cos θ is the equation of the tangent.

In the standard text books on Analytical Geometry the following problem occurs in the exercises on confocals:

The product of the four normals from a point P to the ellipseX2/a2 + Y2/b2 = 1 is equal to where λ1 and λ2 are the parameters of the confocals through P.

The area of the triangle whose sides are a variable tangent to the cycloid, the tangent at the cusp and the tangent at the vertex is maximum when the point of contact of the variable tangent is the point of intersection of the cycloid and the generating circle at the cusp.

When a student first approaches the theory of infinite continued fractions a natural question that suggests itself is how to evaluate the expression

The formulae of interpolation of Lagrange and Newton are easily retained in the memory if one considers a simple way in which each can be derived, for polynomials.

If there are n coins in a bag, of any individual values but of total value P, and if m are drawn at random, without replacement, it may be proved that the mathematical expectation, or probable value, of the draw is mP/n. It must be well known, though I cannot find any reference in the texts, that the same result holds if the coins are replaced after each drawing, or indeed if replacements after certain specified drawings, but not after others, are permitted.

Let 1α + 2α + 3α + …. + nα be denoted by Sα. It is well known that Sα can be expressed as a polynomial in n of degree (α + 1). The expressions for S1, S2, S3 …. can be found in succession by elementary methods, which also give numerous relations such as S3 = 12S2S3 = 7S6 + 5S4. The elegant method which I am about to explain is not original. It is due in essence to the Arabian mathematician Alkarkhi (circa 1000 B.C.).