It is shown in this note how, by a simple geometrical construction, the two fundamental solutions of the equation a cos θ + b sin θ = c (to which ± 2 k π added gives, if required, the general solution) may be determined experimentally, using the protractor, without ambiguity of any kind. The method does not involve any laborious graph-tracing, and is at once simpler and more accurate than the geometrical discussions given in one or two text-books. (See, for example, Loney, Part I., page 141, §130; Levett & Davison, page 138).

Some time ago I received from Dr Muirhead the following theorem:—“If lr, mr, nr (r=1, 2, … 3) are the direction-cosines of the axes of the conicoid

In the ordinary text-books on Algebra there is a lack of suitable examples on Multiplication of Determinants. Most of the examples that are given are particular cases of the theorem

where A1 A2, …, B1, …, are the co-factors of a1, a2, …, b1 …, in D.

Let ABC be the Δ, AD ⊥ to BC produced to meet circumcircle in K, BL a diameter of circumcircle, LM ⊥ to AK. Let BK, KG, CL and LA be joined.

Mr Burgess' pretty solution of the problem “to draw a circle cutting three circles diametrically” suggests a question as to the use of sign in treating of co-axial circles.

These limits may be proved very simply by applying the following theorem in inequalities:—

If n is a positive integer and r a a positive proper fraction for the values 1, 2, 3, … n of r, then

Professor Bryan's article on Curvature, etc., in your last issue (p. 219) ends with a challenge. I wonder whether he would be satisfied with the following reasoning to prove that the intersection of “consecutive” normals and the centre of the circle through these “consecutive” points of a curve have the same point as limiting position.

H being the orthocentre of a triangle ABC, we may call the angles HAC, HBA, HCB = x, y, z respectively, for their sum is 90°.

Now tan x tan z =

∴ tan x tan z + tan z tan y + tan x tan y = 1.