The integration of differentials of the form seems to me to be susceptible of a more methodical mode of treatment than that commonly employed. In the ordinary way of presenting the matter there is little choice left to the student, when such an integration is required of him, between a haphazard, tentative process, and the consultation of a text-book, in which lists of ‘formulae of reduction’ are given.

The theorem that the feet of the perpendiculars drawn to the sides of a triangle from any point in the circumference of the circumscribing circle are collinear is ascribed (Gerg. Ann. iv., p. 250, ca. 1814) by Servois, though not with confident knowledge, to Simson. Baltzer, who gives us this information, refers also to Gerg. Ann. xiv., p. 28, p. 280, and to Poncelet. Fuller details, showing how the question of authority has hitherto stood, will be found in an extract from a letter of Mr Mackay's in Nature, xxx., p. 635. Mr Mackay has further stated that he has not found the property mentioned in any of Simson's published works.

The following mnemonics, with one exception, consist of verses or sentences such that if the number of the letters in each word be written down in the order in which the words occur, the desired value will be obtained.

The attempt to enumerate the possible distinct forms of knots of any order, though unsuccessful as yet, has led me to a number of curious results, some of which may perhaps be new. The general character of the methods employed will be obvious from an inspection of a few simple cases, and any one who has some practice in algebra may extend the results indefinitely.

In this paper is considered the form taken in Polar co-ordinates by the equivalents of certain well known expressions in Cartesian co-ordinates which occur in Elastic Solids and in Hydrodynamics.

The accompanying figure (fig. 29) will enable us to transform from Cartesians to Polars by simple geometry.

This paper is designed to call attention to certain extensions of this method, which do not seem to be generally recognised. Most writers consider the ellipse alone as amenable to orthogonal projection, whereas all the conic sections are so.

With the hyperbola we may use this method in three ways—

(1) By projecting a given diameter in the figure to be a transverse axis of the projected figure, when the result will follow by symmetry.

(2) By projecting the hyperbola into a rectangular one.

(3) By an imaginary orthogonal projection from the circle.

This paper included an analytical proof of the following theorem:

If five spheres 1, 2, 3, 4, 5, pass through the same point, and if the four points in which the four spheres 1, 2, 3, 4, intersect in sets of three be coplanar; and if the same be true for the sets 1, 2, 3, 5; 1, 2, 4, 5; 1, 3, 4, 5; it will also be true for the remaining set 2, 3, 4, 5.

The same theorem is true for similar and similarly situated quadric surfaces.

The communication on this subject, as originally made to the society, consisted of a series of theorems, giving (1) expressions for the radii of a great many sets of circles, (2) identities connecting several sets of these radii, and (3) miscellaneous identities closely related thereto. As, however, the paper culminated in a general theorem which may be looked upon as fundamental, and the proof of which makes evident the mode of arriving at the said expressions for radii, and as the relations connecting sets of radii are easily found when attention has been directed to their existence, I have thought it best to print little more than the fundamental theorem and a few auxiliary notes.