Lorsqu' on intègre successivement les fonctions

où le cosinus de l'arc α est multiplié par une puissance à exposant entier de l'arc lui-même,

on reconnaît que les intégrales sont toutes comprises dans la formule

où P et Q représentent des polynomes entiers en α que l'on détermine dans chaque cas particulier.

The properties of the functions Pn, Qn may also be determined very briefly as follows:—

Let u be a function of x; then by integration by parts we have

Let ABCD be a quadrilateral inscribed in a circle (centre O, radius ρ) whose diagonals AC, BD intersect at right angles in S. From S draw SE, SF, SG, SH perpendiculars on AB, BC, CD, DA respectively.

The infinite products for sinx and cosx are most conveniently obtained in a rigorous way from the well-known factorial expressions for sinnθ and cosnθ which, when n is an even integer, take the forms

θ being put equal to and n made infinitely great.

The use of inverse operators is only justifiable when it is obvious what direct operation of the calculus they symbolise.

The purpose of this note is to point out how the usual method of obtaining the integral of this differential equation can be shown as the result of direct operations.

If ABC, A′B′C′ are any two equilateral triangles in a plane, their vertices being taken in the same sense of rotation, of the three lines AA′, BB′, CC′, the sum of any two is not less than the third.

Let ABC, A′B′C′ (Fig. 4) be two triangles equiangular in the same sense. Let BC, B′C′ meet in X. Describe circles round BXB′, CXC′ to meet again in O. Then it is easy to see that the triangles BOC, COA, AOB are equiangular in the same sense to the triangles B′OC′, C′OA′, A′OB′ respectively. Hence the triangles AOA′, BOB′, COC′ are similar;

∴ a. AA′, b. BB′, c. CC′ are proportional to a. AO, b. BO, c. CO, where a, b, c are the sides of the triangle ABC.

The object of this paper is to introduce certain functions analogous to the circular functions. The functions will be denoted by

Formulæ analogous to

will be obtained, and the use of the functions in symbolical solutions of certain differential equations exemplified. The connection of the functions with generalised Bessel-functions of order half an odd integer will be shown.

What I intend to put before you first is a graphical view of the solution of indeterminate equations of the third degree.

The general linear matrix equation has been investigated by Sylvester in a series of interesting papers published in the Comptes Rendus. His method, however, does not show that the solution can be exhibited as an analytical function of the coefficients. The object of this note is to supply such a solution.

Two features characterise the treatment of Geometry as presented in Euclid's “Elements”: (1) the propositions are arranged in a definite sequence which cannot be greatly altered without invalidating the proofs; (2) there are no methods of proof applicable to a large number of propositions. If we except the method of reductio ad absurdum, it is scarcely an exaggeration to say, for example, that in Book I no three propositions are proved by the same method.

Several textbooks of Elementary Geometry have recently been put on the market, and in nearly all that I have examined (and I have gone carefully through many of them) the treatment of tangents is based on what the writers call the Method of Limits. The usual form given to the proof that the tangent at any point of a circle is at right angles to the radius to the point of contact is somewhat as follows.

Amongst the “technical terms” that have come into use in connection with Coordinate Geometry, not the least convenient is the word Power. The only definition of a general kind for this term that I have met with is the following:

“Def.—The result of substituting the coordinates of any point in the equation of any line or curve is called the Power of that point with respect to the line or curve.

“[This definition, first given by Steiner, is now employed by all the French and German writers.]”

“To draw through a given point a transversal of a given triangle so that the segments of the transversal may be in a given ratio.”

Let μ ≡ refractive index of the prism.

Let α, α′, β′, β be the successive angles of incidence and refraction of a ray, in a plane perpendicular to the edge of the prism.

Among the many formulæ which show special relations existing between the circular functions and the Bessel-Function Jn(x), when n is half an odd integer, there is one due to Lommel