Prof. W. P. Milne has cited (Proc. Edin. Math. Soc., Vol. XXXI., p. 90) pairs of nonagons inscribed in a plane cubic curve, and in nonuple perspective. A part from this, and the familiar case of triangles, little attention appears to have been given to polygons in multiple perspective.

The cam problem may be stated as follows:

To design a curve rotating about O so as to actuate by contact a given follower curve to rotate about O′ in a prescribed periodic manner.

Reference may be made to Barr, Kinematics of Machinery, Chapter 5, for the engineer's method of constructing cams by means of a template. A fuller analysis of the problem is aimed at in this paper.

Asymptotic Expressions for the Bessel Functions.

From the asymptotic expansion for Ku(z) it follows that, if − π < amp z < π,

This theorem is also true if amp z = ± π; to prove this consider the formula

The polynomials which satisfy linear differential equations of the second order and of the hypergeometric type have been the object of extensive work, and very few properties of them remain now hidden; the student who seeks in that direction a subject for research is compelled to look, not after these functions themselves but after generalisations of them. Among these may be set in first place the polynomials connected with a differential equation of the third order and of the extended hypergeometric type, of which a general theory has been given by Goursat. The number of such polynomials of which properties have been studied in particular is rather small; in fact, Appell's polynomials

and Pincherle's polynomials, arising from the expansions

are, so far as I know, the only well-known ones. To show what can be done in these ways, I shall briefly give the definition and principal properties of some polynomials analogous to Pincherle's and of some allied functions.

Brill and Noether in their Report, Die Entwicklung der Theorie der Algebraischen Functionen in äerer und nlteuerer Zeit, Abschnitt I., § 16, state in their remarks on Taylor's Theorem that “in its modern form the Theorem appears only as a Corollary (Prop. VII., Cor. II.) and is left without any application.”

1. In obtaining a solution of the differential equations corresponding to the motion of a particle about a position of equilibrium, it is usual to express the displacements in terms of a series of periodic terms, each sine or cosine having for its coefficient a series of powers of small quantities. Korteweg has discussed the general form of such solutions, and, from the developments in series which he has obtained, has deduced certain features of interest. In particular, he has shown that, under certain circumstances, it is possible that certain vibrations of higher order, which are normally of small intensity compared with the principal vibrations, may acquire an abnormally large intensity. Considering the oscillations of a dynamical system having a number of degrees of freedom, and supposing to be the frequencies corresponding to infinitesimal oscillations in the different normal coordinates, Korteweg has shown that these cases of interest arise only when

is zero or very small, where p1, p2,… are small integers, positive or negative; the most important cases occur when

1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equation

and Pn(x) satisfies the differential equation

1. If I is an inflexion on a non-singular plane cubic curve, a variable line IPP′ establishes a (1, 1) correspondence between points P, P′ on the curve. This correspondence defines a perspective transformation of the whole plane, with I for pole, and the harmonic polar of I for axis, of perspective; for, when I is projected to infinity on the y-axis, and its harmonic polar taken for x-axis, the resulting equation

indicates a curve symmetrical with respect to the latter.