In his New Pathways in Science (p. 121) Sir Arthur Eddington propounds the following problem

“If A, B, C, D each speak the truth once in three times (independently), and A affirms that B denies that C declares that D is a liar, what is the probability that D is speaking the truth ?” and he has dealt with it further in the Mathematical Gazette for October 1935.

The problem is “A affirmed that B denied that C declared that D was a liar. If A, B, C and D each speak the truth (independently) once in three times, what is the probability that D was speaking the truth ?”

This problem, first propounded, I believe, by Sir A. Eddington, has aroused considerable interest, and a solution of it was given by Sir A. Eddington in the Gazette, October 1935. His method of attack was, however, somewhat unusual, and the object of this contribution is to show how the problem may be attacked on normal lines, and a perfectly general solution obtained. For this purpose let us suppose that the chance that A speaks the truth is a, that B speaks the truth is b, and similarly with C and D.

The general problem. This paper deals with particular cases of the following problem: A rectangular array of square cells is given; a particle P moves from one cell to another in such a way that, when it is in any cell, it is equally likely to pass across any of its edges. P is liberated in any given cell, and it is required to find the probability that it will ultimately emerge across any stated edge in the boundary of the array.

The design shown in Fig. 1 is unusual in the amount of pattern that

goes to one repeat, only one complete repeat being included in the illustration. It is also unusual in that it includes fifteen-rayed stars.

The relation between a curve and its involute suggests the idea of a pair of curves such that the binormals of one cut the other orthogonally. Let r be the position vector of the current point P on the first curve C and t, n, b the unit vectors in the directions of the tangent, the principal normal and the binormal at P These are functions of the arc-length s of the curve. Since the binormals of C intersect the second curve C1 the position vector of the point P1 of the second curve which lies on the binormal at P is given by

r1= r + ab,.....(1)

in which a is a scalar.

The three-dimensional analogue of Ptolemy’s theorem (Simson’s Euclid, Prop. D) has been given by Cayley (Collected Works, I, 1). The following note contains the theory of such generalizations to space of any number of dimensions.

Taking Euclidean space of n dimensions we denote the Cartesian coordinates of a point As by (xs,a) where the Greek suffix is a dummy and takes the values 1, 2, 3, ..., n, n + 1. We make the convention that xs, n+1 = 1 for every point As.

The Algebra Report is silent on the teaching of indices. The writer has found that some graphical illustration on the lines (probably well known) suggested below gives the work more meaning for boys of various degrees of ability

The first step is to plot the values of 2x, 1·5x, 1·2x, 1x, and 0·5x for the values x = 1, 2 and 3, and, assuming that we may, to join up the sets of points by smooth curves. This, of course, is done rather tentatively, as three points do not give much definition to a curve, but the values of all except 2x when x = 4 can be added and make the curves more definite. To include 24 makes the scale of the graph undesirably small.

If a particle is constrained to move on a sphere, the projection of its acceleration at P upon the tangent plane at P is here called its tangential acceleration. The line of action of the tangential acceleration touches a great circle; if O is a point on this great circle, the tangential acceleration at P is said to be directed to O. A tangential acceleration which is always directed to a fixed point O is called a central tangential acceleration.

The method is best introduced by a simple example.

To work outas a recurring decimal.

Instead of long division by 19, proceed as follows

[Explanation. Divide 2 into 1·0 and write the quotient 5 one decimal place further on than usual.