An identity involving symmetric functions of n letters may in a certain class of cases be extended immediately to a greater number of letters.

For example, the theorem

may be written

and in the latter form it is true for any number of letters.

This celebrated problem—to describe a circle to touch three given circles—admits of a simple experimental solution which might be found interesting by a mathematical class.

The surface on which the circles lie may be either a plane or a sphere, but the latter case is the more convenient to deal with, and will be the one taken.

§ 1. Two permutations of the natural order (123 … n) are said to be conjugate when each number and the number of the place in the one permutation are interchanged in the case of the other permutation.

The theorem in question applies to a function f(x) which, together with its derivatives, is continuous in the neighbourhood of the maximum or minimum value.

The following two propositions are converses of Euclid III., 11 and 12.

Prop. 1. If the circumferences of two circles whose centres are A and B have a common point C lying in the straight line AB, then the circles touch each other externally at C.

An important note in the Proc. Edin. Math. Soc. of May 1904 pointed out that the then prevalent School method of tangents made it difficult for the pupil to gain sound knowledge in his later studies. The same criticism may, I submit, be directed against the treatment, now prevalent in school texts, of (1) surface of sphere, (2) volume of sphere, and (3) volume of cone and pyramid.

The array of binomial coefficients written in triangular form has many curious properties, particularly so if alternate rows or alternate columns receive a negative sign. We may consider such a scheme, developed from its top right hand corner, containing n rows and columns.

The values of the principal radii of curvature ρ1; ρ2 at a point on an ellipsoid are α2/p, β2/p where α2, β2 are the squares of the semi-axes of the central section parallel to the tangent plane at the point, and p is the distance from the centre to the tangent plane.

The figure shews the dissection in the form

Let m, x denote AD, DE, where E is the projection of A on BC.

Quadrilateral 1 is constructed with sides ½ (c + x), ½ (c – x), ½ (m + ½a + x), ½ (m – ½a – x); quadrilateral 6 with sides ½ (b – x), ½ (b + x), ½ (m – ½a – x), ½ (m + ½a – x). The square on AB is then dissected into four quadrilaterals equal to 1, plus the shaded square (AD2); similarly for the square on AC. These quadrilaterals are reassembled in the squares on BE, EC respectively with the right-angled corners in the reversed positions, so as to enclose squares each equal to DE2. Finally two quadrilaterals are subdivided (4, 11) and (3, 5, 10, 12) to fit in as shewn in the figure.

The two standard formulae required are used in the forms

The values corresponding to 0°, 30°, 45°, 60°, 90° are assumed known.