If we write the product

in the form

the coefficients have been called by Professor Nielsen (following Thiele) the Stirling Numbers of the First Species, because James Stirling, in his Methodus Differentialis (1730), was the first writer to draw attention to their use, and furnished a small table of their initial values.

The complete elliptic integrals K and E are functions of their modulus k which satisfy the equations

respectively.

In his notable memoir on the Bessel Functions, Sonine proves the elegant result that

m > –½, provided a plane triangle can be drawn with sides a, b, c; otherwise the value of the integral is zero.

In the geometry of the plane the logical interrelations of figures may often be rendered clearer by considering the plane to be a part of space of three dimensions. Thus, by taking the plane figure as part of a more extensive configuration in space of three dimensions, the elucidation of its properties, and in particular its relation with other figures, are often facilitated. Similarly, the figures of space of three dimensions can sometimes be treated more advantageously and compendiously by considering them as parts of figures in a space of four dimensions, and so on. As a single instance we may take Segre's elegant and powerful mode of treatment of the quartic surface which possesses a nodal conic. This surface he obtains as a projection in space of four dimensions of the quartic surface which constitutes the base of a pencil of quadratic varieties. In the following paper this mode of treatment has been applied to the interesting variety of the Cremona transformation in the plane known as the De Jonquieres transformation, a transformation which possesses some intrinsic interest in view of the fundamental rôle which it plays in the theory of Cremona Transformations. By the aid of a surface in space of three dimensions, a variety in space of four dimensions, etc., simple constructions are given for the De Jonquières transformation between two planes, between two spaces of three dimensions, etc., respectively.

This paper deals with a system of tetrahedra in a sphere corresponding to the co-symmedian system of triangles in a circle. Such a system of tetrahedra, so far as the writer knows, has not been hitherto discussed. The condition that a tetrahedron may have a symmedian point is given in Wolstenholme's “Problems” (1878).

The discovery of the configuration known as a Determinantal System of Points was made by Dr W. P. Milne while investigating the problem of the “Generation of a cubic curve by apolar pencils of lines.” In the Proc. L. M. S., Ser. 2, Vol. 15, Part 4, he obtained a complete solution, but in unsymmetrical form. He therefore suggested to me that as he was up to that time unable to find a symmetrical solution for the general cubic I should investigate Determinantal Systems for the case of rational curves. The results of my investigations are given in this and the ensuing paper, and have enabled Dr Milne to solve the general problem in symmetrical form in a paper which will appear in the Proc. L. M. S. at an early date. I commence from the result given in the paper by Dr Milne, entitled Determinantal Systems of Points, in the Proc. E. M. S., Vol XXXIV., Part 2.

It has already been shown that nine points can be found on a conic to form six triads all apolar to a given triad (ABC), the hessian lines of these six triads and of ABC being concurrent.

The construction of the corresponding system in the twisted cubic is easy. In the first place, the poles of the six triads P1Q2R3, etc., will lie on the plane of the triad ABC to which they are all apolar. Secondly, the hessian lines of these triads, which in the conic were concurrent, will now be generating lines of a quadric circumscribing the twisted cubic. Now the pole of a triad is in the osculating plane at any of the points of the triad, hence the osculating plane at Pl will contain the poles of the triads P1Q2R3 and P1Q2R3

Dr E. M. Horsburgh has given the approximate expression for the length of an arc of the catenary:

where C is the length of the chord and T the sum of the tangents at its extremities, analogous to Huygens' approximation to the length of a circular arc:

where C′ is the sum of the chords of the two halves of the arc. These approximations are not confined to these particular curves, but hold, within wider or narrower limits, for any continuous curve. In fact, if s is regarded as a small quantity of the first order, as are also C, C′ and T, the difference between s and either of the quantities ⅓(2C + T) and ⅓(4C″ – C) is a small quantity of the fifth order. Other expressions of a similar form can be found involving the chords and tangents of the half-arcs.

The following discussion of the analytical continuation of the hypergeometric function is believed by the writer to possess the advantages of brevity and simplicity.

Denote the integrals of Gauss's Equation

which are regular near 0,1, ∞, respectively by

In 1909 Dr Thomas Muir, in a paper on the above topic, gave several theorems involving the derivation of a circulant, and it is the writer's purpose in this paper to extend these investigations with a number of other results.