Let ABC be the Δ, AD and BE ⊥s to BC and AC, CK and BM ⊥s to ED.

Let the three circles have centres, A, B, C, and let their radical centre be T, and the

circumcentre of ΔABC be 0. Join TO and produce it to T1, so that T1O = TO. T1 is the centre of the required circle. Join T1A and T1C, and draw diameters A1^AA2, and C1CC2 perpendicular to T1A and T1C. ⊙ with centre T1 and radius T1A2 passes through A1.

The following note shows how the conditions for the reality or equality of the roots of a cubic can be obtained from the similar conditions for a quadratic. The method does not involve the use of the calculus or the properties of turning points, nor of the imaginary cube roots of unity.

If ABDC be a parallelogram, of which AD is a diagonal, and if 0 be any point which, by joining

to A, B, C, D, makes three triangles, OAB, OAD, OAC, each of these being taken as positive in sign so long as O lies within the region selected (i.e. within ABD), then:

ΔOAC – ΔOAB = ΔOAD.

There are many well-known graphical solutions of the quadratic equation, in which the roots are obtained as the intersections of a circle with a straight line. The practical objection to these solutions is that a fresh diagram has to be drawn for each equation that is to be solved: and the time occupied in constructing the diagram is greater than the time required to solve the equation by the ordinary arithmetical method.

In Mathematical Notes for May 1915, Dr J. M'Whan gives a note on the determination of centres of curvature, stating that “the only assumption involved is, therefore, as stated, that of the definition of the centre of curvature as the limiting position of the intersection of these normals.”

The following note gives a proof of the theorem that if x, y, z are any three values of u which satisfy the differential equation where l, m, n are constants. If the dots denote differentiation with respect to t, we have

The following is a direct proof that any n + 1 particular integrals of the differential equation

where

and P1, P2 … Pn are functions of t alone, are connected by a linear relation.