Page 52, par. 9, line 2.

“FP′F1P” should read “FP′f1P.”

Page 53, par. 14, line 11, insert after the word parabola

“indicated in Fig. 16 by RR1R′R′1 which also show that the parabola is the upper limit of the ellipse or lower limit of the hyperbola”

also for the word “and” substitute “then” in same line.

Page 54, par. 16, line 8, for “f1S2f2” read “f1xf2” and in line 9, for “f2Sf4” read “f2S2f4.”

Page 55, par 19, line 3, for “f1f2” read “f1f4,” and in line 4. for

“ca′1c1 and in line 5, for “xa1Q1C1” read “xa1Q1c”

in par. 20, same page, line 2, for “xS bisects P′S1Q′” read “xS1, bisects angle P′S1Q′”

same par., line 8, for “direction” read “directrix.”

Page 56, par. 24, line 1, for “CY and C1Y” read “cy and c1y,” and in line 3, for “YF” read ”yF,” par. 25, line 5, for “ScSx” read “SCSX,” and line 6, for “Sx” read “Sk,” and for “circle” read “radius.”

In a plane, a point O and a straight line OH drawn through O are given. OH is the bisector of an unknown angle YOX, which it is required to determine by the following conditions:

A point I, given by position in the plane of the figure, is connected with the straight line OH by the given angle IOH =θ, and by the distance OI=c, from the point I to the vertex of the angle. This point I is the middle of a chord AB inscribed in the angle YOX, Furthermore the product OA × OB of the distances to the point O of the extremities of this chord is equal to a given quantity K2.

The following is an elementary discussion of certain propositions in solid geometry which are commonly left to a later stage.

Let ABCD be a quadrilateral formed of rods freely jointed, let points E, F, G, H be taken on AB, BC, CD, DA respectively and let EG, FH be joined by rods freely jointed to the former. Let us investigate whether the resulting framework is rigid.

The following arrangement of the proof of this theorem could, I think, be given at a comparatively early stage, even if the necessary case of De Moivre's theorem had to be proved as an introductory lemma.

Let u, v be two rational integral algebraic functions of x, y with real coefficients, and let c be a simple closed contour in the plane. As the point (x, y) travels round c let those changes in the sign of u that take place when v is positive be marked and let (u, v; c) denote the excess in number among these of changes from + to − over changes from − to + *.

Let φ(a b′ c″) denote the compound determinant,

Then if A, B, etc., denote the co-factors of the elements a, b, etc. in the determinant (a b′ c″), we have

Definition. When two conics intersect each other at two points in such a manner that the tangents and normals of the one become the normals and tangents of the other, they may be said to cut each other orthogonally.

Theorem. 1st, A given conic can be cut at every point on it by two conics which are orthogonal to it; 2nd, every conic orthogonal to a given conic passes through two fixed points on the axis of the given conic.

Dygogram is the name of a curve, invented by Archibald Smith and employed in his Admiralty Manual of the Deviation of the Compass, to give a graphic representation of the varying magnetic field of a compass as the ship is swung round in azimuth; a description is given of the Dygogram by Maxwell in Electricity and Magnetism, §441.

The method here explained affords a complete solution of the problem to determine by geometric construction the values of any number of unknowns connected by an equal number of equations of the first degree. The construction consists entirely of straight lines, and can be carried out by the aid of a straight-edge and a scale; or, in its modified form, in which parallel lines are required, by these instruments along with a set-square.

When P is joined to four points A, B, C, D coplanar with P, a pencil of four lines is formed whose cross ratio is constant if ABCD are collinear. If A, B, C, D are not in a line the cross ratio P(ABCD) has a value which in general varies with the position of P, but which should be known when P is given in position and also A, B, C, D. A simple expression for the cross ratio is given and its utility in locus problems is illustrated by a variety of simple examples, which in several cases furnish methods for constructing a general cubic curve, with or without double point, a trinodal quartic, etc.

In the Proceedings of 1905–6 Mr Pinkerton gave an extension of the nine point circle to a nine point conic. This raises the question of the extension of the geometry of the circle and triangle to that of the conic and triangle. If a triangle with its associated system of lines and circles be orthogonally projected on a second plane we have a triangle with an associated system of lines and homothetic ellipses. Pairs of perpendicular lines are projected into lines parallel to pairs of conjugate diameters. Such lines will be called, for shortness, in the sequel, conjugate lines. In any relation between lengths of lines, these lengths will be replaced by their ratios to the lengths of the parallel radii of one of the homothetic ellipses.

In a paper under the above title in Vol. XXIV. of the Proceedings it is shown that in a certain system of co-ordinates, the equation of the first degree represents a circle orthogonal to a fixed circle. It follows that any purely graphical theorem regarding right lines in a plane can be extended to orthogonals to a circle. This may be seen otherwise by projecting the figure of right lines on a sphere, the right lines thus becoming circles orthogonal to a circle on the sphere; and then inverting the sphere into the original plane. The geometrical method shows that the extension may also be applied to theorems involving one circle as well as right lines, the circle remaining unchanged, while the lines become orthogonals to a circle; the Pole and Polar Theorem, Pascal's and Brianchon's Theorems are examples. But plane figures involving more than one circle cannot in general be transformed in this way. We cannot, for instance, deduce the construction for a circle touching three great circles on a sphere from the known construction for a circle touching three lines in a plane; nor the Gergonne construction for circles on a sphere from the corresponding method in a plane.

In this paper the factorization of arithmetical numbers of the form , where x; is a rational number such that kx is a perfect square, is investigated by means of a trigonometrical transformation. The number k will be taken to be prime for the present.

Any property of a curve which depends solely upon its form may be styled intrinsic. Thus the circle of curvature at a point, the relation between s and ψ, the envelope of the normals, the envelope of straight lines making a constant angle with the curve, etc., are all intrinsic properties. It is proposed to investigate a few of these properties in this paper.

Being given a fixed line (the directrix) and two fixed points S, s (Fig. 15), then, if z and Z are two points on the directrix, the lines zp and ZP are said to correspond if zp be parallel to SZ and 8zsz be parallel to ZP.

Theorem 1. If pairs of corresponding lines meet in p and P respectively, then sp is parallel to SP ; and, if any line goes through p, the corresponding line goes through P.

One of the well known properties of the number 7 is that when is reduced to a decimal, the periods of two digits are obtained to infinity by successive doubling. It is interesting to find for what other numbers this property is true.

Let n be any number, and r the base of notation, then