The expansion of the symmetric determinant

is familiar. It is of six terms; but one term, fgh, is duplicated, and so there are five distinct terms. If we ascertain the number un of distinct terms in the expansions of symmetric determinants of increasing order, we find the following sequence of values, for n = 0, 1, 2, …,

where by convention u0 = 1.

The Method.—An equation F(x) = 0 has, a root x = r, not known exactly. From a first approximation to r, x = a, a second approximation, x = b, is obtained from the formula

From b a third approximation, x = c, is obtained by the same formula, and so on. The method is pointless unless the successive approximations do actually tend to r; a rule that ensures this is due to Fourier.

In the theory of statistics a set of quantities a1, a2, …, av is considered, and called a distribution. The moments of this distribution about its origin are defined by the equations

.

Morley's theorem states that if ABC be any triangle, and if those trisectors of the angles B and C adjacent to BC meet in L, and M, N be similarly constructed, then the triangle LMN is equilateral.

In this note an expansion is found for the expression xn + yn in terms of x + y and of xy. Two illustrative applications are appended.

The integrals referred to here are those given by the following

Theorem:

where the integration is extended over

and l > 0, m > 0, n > 0, a ≤ a ≤ b < ∞.

The usual proofs of Desargues Theorem employ either metrical or analytical methods of projection from a point outside the plane; and if it is attempted to translate the analytical proof by the von Stuadt-Reye methods, the result is very long and there is trouble with coincidences. It is the object of this note to give a short geometrical proof which, in addition to the usual axioms of incidence and extension, uses only the assumption that a projectivity which leaves three points on a line unchanged also leaves all points on it unchanged. Degenerate cases are excluded as having no interest.

If P1, P2, P3, P4 are four points on a circle C, and P234 is the orthocentre of triangle P2 P3 P4, P134 the orthocentre of triangle P1 P3 P4 and so on, then the quadrilateral P234 P134 P124 P123 is congruent to the quadrilateral P1 P2 P3 P4. This theorem seems to be due to Steiner (Ges. Werke, 1, p. 128; see H. F. Baker, Introduction to Plane Geometry, 1943, p. 332) and has appeared frequently since in collections of riders on the elementary circle theorems.