The theorem in question is the following:—if φ(x) is a function that either always increases or else always decreases as x increases from a to b, then

where a < ζ < b

Parallel to sides.

(a) Let O and 0' be two reciprocal points,

DE(α), FG(β), HI(γ) the intercepts made on the sides by lines drawn parallel to the sides through O;

D'E'(α'), F'G' (β'), H'I' (γ') similar intercepts for O'.

Take first the case of a surface S rolling on a plane, the instantaneous axis of rotation being a line in the tangent plane at the point of contact. Take that line as x-axis, and the normal as z-axis, and let the equation to the surface be

Then if p is small, the equation of the section of the surface S by the plane z = p is

Now for rolling of this sort it is clear that the successive axes of rotation during a short time will be generators parallel to OX of a cylindrical surface which will touch S in the neighbourhood of O.

If, in any triangle ABC, the angle A > the angle B, and lines AD and BE be drawn so that the angles BAD and DAC are, respectively, greater than the angles ABE and EBC, then is BE > AD. (Fig. 6.)

Proof: Draw AD′ and AD″ making angles BAD′ and D″AC, respectively, equal to the angles ABE and EBC.

The theorem is

If the straight lines bisecting the angles at the base of a triangle and terminated by the opposite sides be equal, the triangle is isosceles.

This theorem was in the year 1840 communicated by Professor Lehmus of Berlin to Jacob Steiner with a request for a pure geometrical proof of it. The request was complied with at the time, but Steiner's proof was not published till some years later.

Fourier's Series were first applied to the solution of the Differential Equations which occur in the Theory of the Conduction of Heat and that of the Vibrations of Stretched Strings, namely,

In the first, when the flow of heat is linear and the initial temperature is given in the range 0 < x < π by f(x), the solution can be put in the form

In this case there is no difficulty with regard to the differentiation of the infinite series, term by term, as the factor causes the series which we thus obtain to remain uniformly convergent with respect to x and t in the intervals concerned.

The proof given in Chrystal's Algebra, II., pp. 42–5, of this very important theorem is deduced from elementary algebraical principles : and, though somewhat involved, is of great value, as it establishes what must be considered a fundamental theorem in the Calculus.

The property is, that if such a decimal has 2k repeating figures, the first k are a1, a2, &c, in order, and the second k are b1, b2, &c, in order, and the a'a and b's are such that

for example,

Menelaus's theorem is

If a transversal DEF cut the sides of triangle ABC at the points D B F, then

and conversely.

I have recently shown that any two straight lines that are isotomically conjugate with reference to a triangle are the asymptotes of a conic circumscribed about the triangle; and that, conversely, any circumscribed conic has for its asymptotes two such lines. Now there is a one-fold infinity of conies having a given pair of straight lines for asymptotes; and, as there is a two-fold infinity of pairs of isotomic lines in the plane (with reference to a given triangle), it follows that two conditions must subsist among the coefficients in the equation of a conic, in order that its asymptotes may be isotomic lines.

In a paper published in 1897 it was suggested that the Law of Extensible Minors was applicable to Kronecker's linear relation between the n-line minors of an axisymmetric determinant of the (2n)th order, the concluding sentence of the passage being— “For example, knowing from Kronecker that

if the terms be minors of the axisymmetric determinant

we can at once vouch for the identity

in connection with the axisymmetric determinant

No proof was given, as the matter turned up quite incidentally, and verification of the cases made use of was all that was necessary.

Just over sixty years ago—in December 1841—a Commission on Weights and Measures made the following proposals towards the establishment of a decimal coinage in this country: (1) the sovereign to be the unit; (2) a coin worth two shillings to be introduced under a distinct name; (3) a coin equal to the hundredth part of a pound to be established; (4) the farthing to be considered as the thousandth part of a pound; (5) other coins bearing a simple relation to these (including the shilling and sixpence) to be circulated.

The remark of Carnot quoted by Dr Mackay at the last meeting, regarding the mutual dependence of the four equations derivable from the same figure in the case of each of these theorems, led me to consider more fully the mutual dependence of the equations in question; and the following are some of the results arrived at.

Let ABC be a triangle and AD the bisector of the angle A meeting BC in D. (Fig. 8.)

For simplicity let us suppose AB > AC.

From AB cut off AE = AC, and join DE. Then by Euclid I. 4 it follows that the triangles AED, ACD are equal.

This proof of VI. 3 has the merit of depending only on I. 4 and VI. 1. A similar proof for VI. A can be got by cutting off AE′ = AC from BA produced.

In Figure 11 let ABCD be a contraparallelogram having AB = CD, AD = BC.

Let O, P, Q, R be the midpoints of AB, BC, AD, DC respectively. They obviously lie in the line parallel to AC and to BD, and equidistant from them.

Let U be the mid-point of AC and V that of BD.

OURV is a rhombus each of whose sides is half of AD or BC, and parallel to AD or BC. PUQV is a rhombus each of whose sides is half of AB or CD, and parallel to AB or CD.

Let P1 and P1′ be images of each other with respect to a circle of radius a. (Fig. 13.) Draw AOB the diameter perpendicular to OP1′P1. If r1,r1′ represent respectively the distances of P1, and P1′ from 0, the centre of the circle, the condition is r1r1′ = a2.