In view of the reduction of other facilities by the war, the maintenance of the Library service seemed so desirable that the risks of loss though they could not be ignored must be taken. To reduce them, each volume issued since the outbreak has had a jacket bearing the name of the Association and an address, and a request, to which few borrowers pay any attention, for a monthly notification if the volume is being retained.

A Set of numbers aij (called elements) arranged in a rectangular array of m rows and n columns constitutes a matrix of orders m × n. For example, of the arrays

the first is a matrix of two rows and three columns, i.e. of orders 2 × 3, the second is a square matrix of orders 3 × 3.

From a square matrix of orders n × n we can form a determinant of order n. Thus from the above square matrix we can form the determinant of order 3

The rectangle ABCD is supposed to lie in a vertical plane, with AB horizontal. P is any point in the horizontal plane through CD. Prove that ∠APB < ∠CPD.

Provided that the perpendicular from P on to CD meets CD between C and D, this theorem is true and easily proved by comparing the angles into which a plane through P perpendicular to the rectangle divides the two angles considered. The sine of each of the angles into which ∠APB is divided is less than the sine of the corresponding angle in ∠CPD.

In Routh’s Elementary Rigid Dynamics (Sixth edition, 1897, § 300) a section is given to the application of the principle of linear momentum to determine the gradual changes produced by alterations of mass, with examples; but I cannot recall that any modern textbook explicitly points out that the formulae

M dV/dt = F .....(1)

and d(MV)/dt = F, .....(2)

which serve to determine the motion of a body of constant mass, are not necessarily true when the mass is changing.

In Analytical Geometry there is a most useful equation which is very rarely used and seems to be very unfamiliar to candidates in school examinations. It is the equation of the chord of a conic whose mid-point is (x1, y1). Speaking from a long experience as an examiner, I am always astonished at the unfamiliarity of candidates with this particular equation. If one sets a question asking the candidates to obtain this equation and follows this with a simple rider designed to show its usefulness, even if the candidates know how to find the equation, they rarely solve the rider by means of it.

The following method of introducing the ideas of rotation and strain into the study of Elasticity depends upon the identity

m · n grad u=n · (m grad u + m × curl u).....(1)

and upon the fact that if the straight line PQ undergoes a small rotation ω about an axis through P, the displacement of Q is

Consider first of all a rigid body which undergoes a small displacement. Let the point P of the body move to P′. We shall represent PP′ by u and regard u as typical of a vector field extending throughout the space originally occupied by the body. Let Q be a point near to P and represent by m.

The points A, B, C on the circle x2 + y2=R2 are

(R cos θi, R sin θi) (i = 1, 2, 3).

We may without loss of generality assume 0 ≤ θ1 < θ2 < θ3 < 2π.

If we form the sum S(n) of the digits of a number n, and then the sum of the digits of S(n), and so on, in a finite number of steps we reach a number of one digit, R(n), which we call the reduce of n.

Starting with a number prime to 9, say 2, consider the reduces of powers of 2; we obtain the sequence 2, 4, 8, 7, 5, 1, 2, 4, 8, ..., which recurs after the sixth term. Consider next a five-digit number, say 16427; from this we derive 15794 formed by replacing each digit of 16427 by the reduce of the sum of the remaining digits. Repeating the transformation, we obtain in turn 73184, 72461, 49751, 48137, 16427, the number being restored to its original value after only six transformations. It will be found that a three-digit number repeats after eighteen transformations and yet for a twenty-digit number only six transformations are needed.