through a large lecture-hall, is had by taking a thin wooden board, cut to any chosen shape, with the corner edges of the boundary smoothly rounded, and winding a stout black cord round and round it many times, beginning with one end fixed to any point, I, of the board. If the pressure of the cord 011 the edges were perfectly frictionless, the cord would, at every turn round the border, place itself so as to fulfil the law of equal angles of incidence and reflection, modified in virtue of the thickness of the board. For stability, it would be necessary to fix points of the cord to the board by staples pushed in over it at sufficiently frequent intervals, care being taken that at no point is the cord disturbed from its proper straight line by the staple. [Boards of a considerable variety
Fig- 4.
of shape with cords thus wound on them were shown as illustrations of the lecture.]
§ 36. A very easy way of drawing accurately the path of a particle moving in a plane and reflected from a bounding wall of any shape, provided only that it is not concave externally in any part, is furnished by a somewhat interesting kinematical method illustrated by the accompanying diagram (fig. 4). It is easily realized by using two equal and similar pieces of board, cut to any desired figure, one of them being turned upside down relatively to the other, so that when the two are placed together with corresponding points in contact, each is the image of the other relative to the plane of contact regarded as a mirror. Sufficiently close corresponding points should be accurately marked on the boundaries of the twro figures, and this allows great accuracy to be obtained in the drawing of the free path after each reflection. The diagram
shows consecutive free paths 74*6—32*9 given, and 32*9— 54*7, found by producing 74*6—32*9 through the point of contact. The process involves the exact measurement of the length (/)—say to three significant figures—and its inclination (0) to a chosen line of reference XX'. The summations 2 I cos 20 and 2 I sin 20 give, as explained below, the difference of time-integrals of kinetic energies of component motions parallel and perpendicular respectively to XX', and parallel and perpendicular respectively to KK7, inclined at 45° to XX'. From these differences we find (by a procedure equivalent to that of finding the principal axes of an ellipse) two lines at right angles to one another, such that the time-integrals of the components of velocity parallel to
Fig. 5.
them are respectively greater than and less than those of the components parallel to any other line. [This process was illustrated by models in the lecture.]
§ 37. Virtually the same process as this, applied to the case of a scalene triangle ABC (in which BC = 20 centimetres and the angles A = 97°, B—29°'5, C = 53°‘5), was worked out in the Royal Institution during the fortnight after the lecture, by Mr. Anderson, with very interesting results. The length -of each free path (I). and its inclination to BC (0), reckoned acute or obtuse according to the indications in the diagram (fig, 5), were measured to the nearest millimetre and the nearest integral degree. The first free path was drawn at random, and the continuation, through 599 reflections (in all 600 paths), was drawn in a manner illustrated by fig. 5, which shows, for example, a path PQ on one triangle continued to QR on the other. The tw o when folded together round the line AB show a path PQ, continued on QR after
