In continuation of the Mathematical Gazette, p. 389, September 1914, on Linear Dynamics, the investigation is resumed here of the rolling and pitching of the body of a carriage on the springs, such as experienced in a railway carriage, tram, or motor car, and the effect can be studied as realised ; but most instructively in the motibus, where the action is felt on a more violent scale, vibrating with dynamical interest and poetry.

1. Begin by supposing the carriage body lowered down and deposited on four springs at the corners of a rectangle, sinking down vertically on the springs an equal distance c suppose, the permanent average set of the springs.

A vertical oscillation of the body will then synchronize, as on p. 393, with a simple equivalent pendulum of length c, assuming the Law of the Spring (Hooke's, the Linear Law).

Another function of the pentagon dissection, Fig. 19, is to provide, by means of an additional cut in each pentagon, an eight-part type of dissection of a pair of quadrilaterals. This, which I will call the first type of eight-part dissection of quadrilaterals, is shown in Fig. 24. The dissection of the pair of quadrilaterals is shown in full lines, and dotted lines are added to show a pair of auxiliary pentagons, the lettering being the same as in the previous figures. A dissection of the pair of pentagons is shown, and there is an additional cut in the first pentagon joining T to the middle point X of the base AB, and an additional cut in the second pentagon joining W to the middle point Y of the base CD. The other new lines shown in the figure are merely consequential on the formation of a pair of quadrilaterals by means of these two cuts. There is no relation between the dimensions of the quadrilaterals except the equality of their areas. Each quadrilateral is divided by six lines, two or more of which are broken ; four of these lines are half sides of the other quadrilateral, and the other two are half diagonals, one of one quadrilateral and the other of the other quadrilateral.

§ 11. We are now in a position to understand how Napier completed his Tables.

We take, to begin with, his First Table. The logarithm of the first term is zero. From § 10, Theorem I., the logarithm of the second term lies between r-s and (r-s)r/s; that is, between 1.000 000 0 and 1.000 000 I. If this is taken as the A.M. of these two numbers, we have for the logarithm of the second term in that Table the value 1.000 000 05. The remaining terms in that Table form with these two a G.P., so that the logarithms have a common difference 1.000 000 05, and the logarithm of the last term can be taken as 100.000 005.

I propose to give you in outline the short course on Indices and Logarithms taken at the present time by the Upper Fifth Form at the Holloway County School. Soon I hope it may be attempted by the Lower Fifth Forms, and, later on, perhaps by kind permission of the examining authority, it may be found possible to include it in the work of the Fourth Forms.

§ 1. Index Notation.—Let a denote any positive numerical quantity, and n any positive integer. Then “ x a” means “ multiplied n times in succession by a.” “an” alone means “1 x an”, i.e. “unity multiplied n times in succession by a.” The index n tells us the number of multiplications.