To find the ratio which gives the greatest chance that through a point, chosen at random within a triangle, six lines may be drawn to divide the triangle in that ratio.

A man has three vessels, whose capacities are 3, 5 and 8 pints respectively. The largest is full of water. He desires to divide this water into two equal parts by using these vessels only What are the simplest ways of doing this?

Graphical solution.

Call the vessels X, Y and Z respectively, and the amounts of water they contain x, y and z respectively.

On a récemment signalé des simples constructions qui mènent d’un triangle plan tout à fait arbitraire à un triangle regulier. On peut établir que ce triangle dérivé jouit de cette remarquable qualité en s’inspirant à une profonde maxime d’Abel (” il faut énoncer un problème de manière de le pouvoir toujours résoudre “); on doit alors se proposer en général de déterminer de quelle espèce est le triangle résultant de l’application d’une des constructions citées. Dans plusieurs cas il arrive que l’expression de la longueur d’un quelconque de ses côtés est une fonction symétrique des éléments du triangle primitif ; alors la régularité du triangle d’arrivée est hors de doute. Il est aussi à remarquer qu’à l’expression cherchée on arrive très souvent par un procédé (qui rappelle la “ triangulation géodésique “) qui mérite d’être signalée à tous ceux qui auront à l’avenir à s’occuper de questions analogues à celles étudiées dans le mémoire qui suit.

Dr. C. N Srinivasiengar, in his published papers (7, 8, 9 and 10), and Dr. W E. Deming, in a private letter, gave examples of differential equations for which the so-called complete primitives are not really complete, and yet the supplementary solutions are not singular solutions in the envelope sense. In the ensuing correspondence I expanded a remark of E. B. Wilson (11) “It may happen that the arbitrary constant C enters into the expression F(x, y, C) = 0 in such a way that when C becomes positively infinite (or negatively infinite) the curve F(x, y, C) = 0 approaches a definite limiting position which is a solution of the differential equation, such solutions are called infinite solutions.”

The following paper arose out of an attempt to improve upon the familiar “match puzzles” one sees in the popular press and which are generally too puerile to be interesting. Following some desultory experiments, it developed that the methods of construction postulated are capable of determining all points obtainable with ruler and compass, but no others. The sequence of results below is arranged to establish this conclusion.

Mr. A. W. Siddons (Harrow): As they say at company meetings, I hope that we may take this Report as read, and very good reading it is. I may say at once that I am not here to criticise the Report if by “criticise” we mean “criticise adversely”. There are two points about which I wrote to the July Gazette.

A familiar method of proving theorems in elementary projective geometry is to establish by a geometrical construction the existence of a (1, 1) correspondence between the elements of two simply infinite rational systems (such as ranges of points, or pencils of lines) and to conclude that such a correspondence is projective and to use this result to deduce the theorem desired. The argument underlying this reasoning is that the relation between the coordinates x, y of corresponding elements (x), (y) of the two systems is given by an algebraic equation f(x, y) = 0, and that if this relation is such that to each value of x corresponds one value of y and vice versa, then the relation f(x, y) must reduce to a bilinear equation, of the form Axy + Bx + Cy + D = 0, which is easily shown to define a projectivity between the corresponding elements (x), (y).

Very often it is proved experimentally that two electrified bodies repel or attract each other, and from this fact is deduced the conclusion that there exist two kinds of electricity of such properties that bodies of the same kind of electricity repel and bodies of different kinds attract each other. This proceeding is not quite correct.

Example. If we consider two integral numbers, then their difference is divisible by 3 or not. But it is obviously wrong to deduce from this fact that we can divide the set of all integral numbers into two such classes, that the difference of two numbers belonging to one class is divisible through 3 and the difference of two numbers belonging to different classes is not. It is known that there exist three classes possessing these properties.

Multiply throughout by a2 and write y for ax+ b ; the equation becomes

where H ≡ac − b2, G ≡ a2d − 3abc + 2b3.

Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written

Accordingly the two remaining roots are also real if

But since α satisfies (i),

and so

Hence if (i) has three real roots, G2 +4H3 ≤ 0; and clearly, when G2+4H3 = 0, two roots are numerically equal to and the third to .

1. It is well known that the solution of the ordinary linear differential equation with constant coefficients

(Dn + a1Dn−1 + + an−1D+an)y = F(x), .....(1)

where y, Dy, ..., Dn−1y take arbitrary values y0, y1, .., yn−1

when x = 0, ..... (2)

can be obtained by multiplying (1) by e−px, (p > 0) and integrating with regard to x from 0 to ∞

All school teachers of the world who are interested in the development of the teaching of mathematics in general and of the teaching of geometry in particular know the three stages A, B, c which are prescribed for the teachers of geometry in England. The father of this splendid idea is W C. Fletcher, formerly Chief Inspector of the Board of Education in London. He issued in March 1909, as Circular 711, a famous regulation: Teaching of Geometry and Graphic Algebra in Secondary Schools.

The interchange of salaried appointments between masters and mistresses in British and American schools is arranged in Great Britain by a Committee representative of the English-Speaking Union, the British Federation of University Women, the Incorporated Association of Headmistresses, the Headmasters’ Conference, and the Association of Headmasters, working in conjunction with Committees in the United States.

In connection with the article on Quadratic Residues in the December number of the Gazette, a neat graphical method, which I first put forward in 1913, may be of interest.