I. Soit ABC un triangle inscrit dans une circonférence donnée, qui a pour centre le point O, et pour rayon R la quantité OA. Si d'un point M pris sur la circonférence on abaisse les perpendiculaires ML, MN, MR sur les côtés BC, CA, AB du triangle, les pieds L, N, R de ces perpendiculaires sont situés sur une même droite RN, à laquelle on a donné le nom de pédale du point M; le point M est le point directeur de la pédale RN.

Rather more than twenty years ago, in a note on this subject, it was shown to the Edinburgh Mathematical Society (Proceedings, II., pp. 16–18) that a special form of continuant, viz., one with univarial diagonals, could be expressed by means of a similar continuant of much lower order. A new mode of proving this theorem, which has lately been hit upon, has unexpectedly led to the discovery that the peculiarity in question is not confined to this special form, but characterises continuants of any form whatever.

The object of this paper is to remove the difficulty that arises in giving a general proof by projection methods of this theorem, without in any way interfering with the single-valuedness of the position of a radius vector tracing out angles from a given initial position, when the values of the trigonometrical ratios are given.

Let a cubic function be denoted by

Put and the function takes the form

Consider the graph of this function, represented by

Soit ABCD un rectangle, AC l'une de ses diagonales; si l'on prend un point M sur la diagonale et qu'on mène les parallèles aux côtés, on a deux rectangles AEFD, AGHB équivalents.

Soit en effet AB une force, AD une autre force, AC sera la résultante; et si l'on prend les moments des deux forces par rapport à un point M de la résultante on aura

ce qui démontre le théorème.

This paper deals with a few of the simpler specialisations of the intersections of a plane curve and the envelope of the family to which it belongs. It follows the method adopted by Professor Chrystal in dealing with the p-discriminant of a differential equation of the first order. This method is specially applicable to definite problems; in these it is safer to work out the result than to rely on theory.

Although the following note makes no pretence at novelty so far as the results are concerned, yet the method employed does not seem to occur in the ordinary text-books on Spherical Trigonometry. I have found this process very useful in explaining to beginners how to distinguish between the various possibilities, and I hope it may be of some interest to other teachers.

This Graduation of the Circumference of a Circle is effected by the aid of an instrument called a trisector which I contrived with the view of trisecting an angle by its assistance, but subsequently perceived that it could help to divide an angle into 5 equal angles, and recently discovered that it could contribute towards dividing the circumference of a circle into 360 equal degrees or arcs.

This paper contains two parts:—

I. An endeavour to remove the present confusion in polar diagrams.

II. On the curves derived from a given curve by increasing or diminishing the vectorial angle in a constant ratio.

This well-known theorem, published in Newton's Arithmetica Universalis, may be formulized thus:—

where sm ≡ the sum of the mth powers of the roots of the equation

Of course, when m>n, certain coefficients pn+1, pn+2, … pm will be zero, and the last term that does not vanish will be .

While Newton's Theorem on the Sums of Powers of the Roots of an equation furnishes a set of lineo-linear equations connecting the quantities s1, s2, s3, … and the quantities p1, p2, p3, … Waring gives the solution of these equations by which the s's are expressed in terms of the p's.

Let the points A, B, C be the centres of three given circles, whose radii are a, b, c; and let d, e, f be the distances BC, CA, AB between the centres (Fig. 41). It is required to find the radii of the circles which touch the circles A, B, C.

This bibliographical note was drawn up to accompany Mr Collignon's memoir Recherches sur l'Enveloppe des Pédales des divers points d'une Circonférence par rapport à un triangle inscrit, printed in this volume, p. 2–34; and if I had remembered (as I ought to have done) the very full bibliography given in L'Intermédiaire des Mathématiciens (Vol. 3, p. 166–168, 1896) by Mr Brocard and others, I should not have commenced it. The result, however, has been that several articles on this particular curve, not noted in the Intermédiaire, have been discovered, and I have thought it worth while to print the information thus gained.

The important inequality of which Prof. Chrystal has given so many examples may be called the “Power Inequality.”

There is a simple theorem of the Differential Calculus which is to a general function f(x), what the Power Inequality is to the function xm.

Let α, β, γ be the known trilinear coordinates (actual lengths of perpendiculars) of the centre of a conic inscribed in the triangle of reference ABC.