[20] To this may be added the extremely useful theorem for such problems, that if any directed quantity L, say, characteristic of the motion of a body, be associated with a line or axis Ol, which is changing in direction, it causes a rate of production of the same quantity for a line or axis instantaneously at right angles to Ol, towards which Ol is turning with angular velocity ω, of amount ωL. If M be the amount of the quantity already existing for this latter line or axis, the total rate of growth of the quantity is there M + ωL. For example, a particle moving with uniform speed v in a circle of radius r, has momentum mv along the tangent. But the tangent is turning round as the particle moves with angular speed v⧸r, towards the radius. The rate of growth of momentum towards the centre is therefore mv × v⧸r = mv²⧸r. [21] See Gray's _Lehrbuch der Physik_, s. 278. Vieweg u. Sohn, 1904. [22] Gray, Royal Institution, Friday Evening Discourse, February 1898. [23] See the _Reports of the Committee on Electrical Standards_, edited by Prof. Fleeming Jenkin, F.R.S., Maxwell's _Electricity and Magnetism_, and Gray's _Theory and Practice of Absolute Measurements in Electricity and Magnetism_, Vol. II, Part II. [24] The writer once, on a thick night, in a passenger steamer in the Race of Alderney, when the engines were stopped and soundings were being taken, saw the reel and cord go overboard, nearly taking one of the men with it. A new hank of cord had to be got and bent on a new reel; an operation that took a long time, during which the exact locality of the ship was a matter of uncertainty. Comment is needless! [25] The tuning of a major third, in this way, is described in the paper entitled "Beats on Imperfect Harmonies," published in _Popular Lectures and Addresses_, vol. ii. [26] The writer well remembers meeting a man of some experience in cable work who was on his way to measure the alternating currents in a Jablochkoff candle installation by the aid of an Ayrton and Perry galvanometer with steel needle! INDEX Atlantic cables, 267, 268 Atmospheric electricity, 226 Atoms, size of, 261 Ayrton, W. E., 296 Baltimore lectures, 254-263 Bertrand's theorem of maximum kinetic energy, 158 Bottomley, James Thomson, 311 Bottomley, William, 7 British Association, electrical standards, 244-253 Cambridge University Musical Society, 24 _Cambridge and Dublin Mathematical Journal_, 25, 31, 78 Carnot, Sadi, 77, 101 Carnot's _Théorie Motrice du Feu_, 87, 101, 108 _et seq._ Cauchy, 294 Chasles, 28, 43 Clapeyron, 101, 112 Clausius, 114 _et seq._ College, the old, of Glasgow, 10 Compass, errors of, 273 "Dew-drop," artificial, 290 Dynamical theorems, Thomson's and Bertrand's, 158 _et seq._ Earth, the age of, 196, 229-243 Earth, tidal retardation of, 230