If ρ be the vector of a corner of a square in one system, σ that in a system derived without inversion, we must obviously have

k being the unit-vector perpendicular to the common plane.

Call 2π/17 α, and let

Then from trigonometry

and

Now let

What is here printed contains merely a list of the memoirs and treatises that may perhaps be found useful for one who wishes to trace the progress of the mathematical theory of heat beyond the stage at which Fourier left it. As discussions of the Fourier series and integrals occur in almost every treatise on the Integral Calculus, I have omitted reference to these. Similar considerations have led me to omit references to the discussion of differential equations, except where these specially dealt with the problem of the conduction of heat.

The formula referred is the condition for the coplanarity of the extremities of four coinitial vectors; namely, if α, β, γ, δ, are the vectors, then

(See Kelland and Tait's Quaternions, p. 62.)

In the first place, it may be pointed out that these equations do involve one condition on the four vectors α, β, γ, δ. For, since the vanishing of a vector involves three conditions, and only the three ratios of the four quantities a, b, c, d, are involved, it is always possible to determine these four quantities so that the condition aα + bβ + cγ +dδ = 0 shall be satisfied. But, in general, the equation a + b + c + d = 0 will not be satisfied; and thus one condition is involved.

[At the first meeting of this Session a paper was read on the value of cos 2π/17, which evidently may be made to depend on the solution of x17 – 1 = 0.* The present paper is the outcome of a suggestion then made, that a sketch of Gauss's treatment of the general equation might prove interesting. To give completeness to the subject the necessary theorems on congruences have been prefixed. The convenient notation introduced by Gauss is here adopted; thus, when the difference between a and b is divisible by p, instead of writing a = Mp + b, we may write a ≡ b (mod p), the value of M seldom being of importance. It is evident that if a ≡ b, then na ≡ nb, and an ≡ bn, n being any positive integer, and the same modulus p being understood throughout. Also a/n ≡ b/n provided n be prime to p. Other properties (similar to those of equations) are easily seen, but only the above are needed here.

It is known that a force acting along any line in space may be resolved into six components. In the most commonly employed resolution these are forces along three lines at right angles, and couples round these lines. But the six components may be taken to be forces along the six edges of a tetrahedron. It is the object of what follows to determine these forces.

[The substance of this paper will appear in the second volume of Professor Chrystal's Algebra.]

There are many proofs of the principle of this planimeter, but all that are accessible to me seem a little beyond the grasp of many students who use the instrument. It seems worth while, therefore, to notice the following proof, which, to the best of my knowledge, is new.

The object of this communication is simply to show how a formula may be obtained which will indicate the most economical speed for a steamer in relation to cargo carried and coals consumed on the voyage.

The object of the following paper is to consider the motion of one or more vortices in a compressible fluid, which is rotating as a whole with uniform angular velocity ω about an axis, taken as axis of z. To save space I shall when possible refer for results to a previous paper in the Proceedings, distinguishing the equations of that paper, Vol. V.; pp. 52–59, by the suffix a.

The theorem that any rational symmetric function of n variables x1, x2, … xn is expressible as a rational function of the n elementary symmetric functions, Σx1, Σx1x2, Σx1x2x3, etc., is usually proved by means of the properties of the roots of an equation. It is obvious, however, that the theorem has no necessary connection with the properties of equations; and the object of this paper is to give an elementary proof of the theorem, based solely on the definition of a symmetric function.

The two problems are :—

1. To divide a given straight line internally and externally so that the ratio between its segments may be equal to a given ratio.

2. To divide a given straight line internally and externally so that the rectangle under its segments may be equal to a given rectangle.

In the following paper I propose to give a short account of Dr Ernst Kötter's purely geometrical theory of the algebraic plane curves. This theory is developed in a treatise which, in 1886, gained the prize of the Berlin Royal Academy; but the contents of my paper are also partly drawn from a course of lectures delivered by Dr Kötter in the University of Berlin, W.S. 1887–88.

The usual methods of proving the existence of regular polyhedra, as given in Wilson and in Todhunter, appear to most students somewhat difficult. It seemed worth while trying, therefore, whether a simpler or more direct proof could not be obtained. The following note shows how this may be done.

Théorème.— On considère tous les cercles σ passant par deux points fixes, dont l'un c est sur la circonférence d'un cercle donné s, et l'autre d sur une droite donnèe l. Chacun des cercles σ rencontre la droite l en un second point d' et la circonférence s en un second point c′: La droite c′d' passe par un point fixe i de la circonférence s, quel que soit le cercle σ considéré.

In examining some of the lines that occur in connection with the recent geometry of the triangle, the cubic whose equation in trilinear co-ordinates is

incidentally appeared, and it seems worth noting the very large number of special points it passes through. These are the vertices, the mid-points of the sides, the inscribed and escribed centres, the circumcentre, the orthocentre, the centroid and the symmedian points, or fourteen in all.