The problem of the greatest term of a binomial expansion is a favourite one in elementary text-books, and its solution is often difficult to a beginner. The difficulty, at least in the case where the index is negative or fractional, seems to be caused by the fact that a “formula” is provided which gives a value for r, such that the (r + 1)th term is the greatest. Moreover, this formula is not always the same. Sometimes it is sometimes ; and unless the student has a very good memory he is sure sometimes to make mistakes. Elementary mathematics ought not to be a memory exercise. It is a platitude to say that the educational value of the teaching of mathematics lies in its training of the powers of reasoning. This element is eliminated when processes of reasoning are reduced to a rule of thumb. As well might one use “Molesworth” as a text-book of the principles of mechanics.

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.

Let O be the centre of the circumscribing circle of ΔABC, A1 the middle point of BC, and EA1OF the diameter at right angles to BC. Draw AX perpendicular to BC and produce it to meet the circle in K. Let H be the orthocentre of ΔABC; join OH and bisect it in N, the centre of the nine-point circle.

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.

Let A and B be square matrices of the same order, with elements in any field F; it is well known that the characteristic polynomials of AB and BA are the same (see, e.g. C. C. Macduffee, Theory of Matrices, p. 23). The proof of this is easy when one at least of the matrices is non-singular; the object of the following remarks (which are not claimed as original) is to point out that the case | A | = | B | = 0 is just as easy. If one attempts to deduce the result in this case from the result in the non-singular case, unnecessary restrictions on the field F are apt to appear (see e.g., W. V. Parker, American Mathematical Monthly, vol. 60 (1953) p. 316). If one proceeds directly to the general case, no difficulties are encountered.

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 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 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.

The familiar Lemma introduced by Goursat in his proof of Cauchy's theorem suggests the following necessary and sufficient condition for differentiability of a complex function f(z).