thrust along AC is nearly perpendicular to the groove, and tends to jam the sliding piece against one side of the groove. The middle point itself is a dead point, the statically applied force there being perpendicular to the required motion.

93. A machine for compounding two parallel simple harmonic motions with any given ratios of period and of amplitude, has been invented by Mr. A. E. Donkin, and is constructed by Tisley and Company.

There are two vertical axles turning in fixed positions, and carrying cranks whose lengths are made equal to the amplitudes of the two motions which are to be compounded. Toothed wheels of various sizes are provided, which can be fixed on the axles (one on each), and the numbers of their teeth determine the ratio of the periods of the two components. As their axes are fixed, a third toothed wheel with movable axis must be employed to connect them; and by this means, when one of the two fixed axles is turned by hand, the motion is transmitted to the other with the required velocity-ratio. The crank carried by the first axle gives, through the medium of a long connecting-rod, a vibratory movement to the lower end of a lever, the upper end of which moves the pen (a glass tube drawn out to a point). The paper is drawn uniformly, in a direction perpendicular to the movement of the pen, by the revolution of a roller, carrying a toothed wheel, which is driven by a train of wheelwork deriving its motion from the revolution of the

above-mentioned vertical axles. One of these, as above stated, gives a vibratory movement to the pen. The other, by means of its crank and a long connecting-rod, gives a vibratory movement of translation to the frame on which the paper rests. The roller which draws the paper is also carried by this frame, and it is necessary that the toothed wheel on the roller should remain in gear with the pinion which is to drive it, in spite of this vibratory motion (which is parallel to the axes of the toothed wheel and pinion). This object is attained by making the pinion of great length (a long fluted cylinder), so that the toothed wheel can slide along it longitudinally. The speed of the paper and the lengths of the two cranks, as well as their velocity-ratio, can be regulated at pleasure, so as to give any required amplitudes and wave-lengths to the two undulations which are compounded. The figures in Plate I. are slightly reduced from curves traced by this machine. The approximate ratio of the two periods is stated on the left and their rigorous ratio on the right. The amplitudes of the two components are equal in these specimens, but the machine admits of their being varied independently.

94. By means of a bell-crank lever, the motion of the pen above described can be exchanged for a motion in the perpendicular direction, and thus vibrations at right angles to each other can be compounded. The figures in Plate II. are slightly reduced from curves thus drawn upon paper fixed to the frame, and not drawn onwards

by the roller; the change from one curve to the next of its kind being effected by unclamping the toothed wheels and turning one of them through a definite number of teeth before clamping again. The ratios of the periods are indicated in the margin.

The following is a more

complete account of the contents of the Plate.

In all the curves in the first line the ratio of the periods is 2 I, two horizontal vibrations being executed in the same time as one vertical. The first and the last figure in this line are parabolas.

In the second line, the ratio for the first four curves is 32, and for the last two curves 3: 1, three horizontal vibrations being made in the same time as two or one vertical.

In the third line, the ratio is 5: 3 for the first three curves, and 5 4 for the last three; five horizontal vibrations being made in the same time as three or four vertical.

In the fourth line, the ratio is 9: 8 for the first three, and 10:9 for the last three, the number of horizontal vibrations being in each case greater by unity than the number of vertical vibrations.

All these ratios can easily be verified by inspection of the curves. For this purpose the student must count how many times he crosses over the horizontal breadth of the figure, and how many times over its height, in travelling along the curve, from one end of it to the other, if it has ends, or until it brings him back to the

point from which he started, if it be an endless curve. In the latter case it is convenient to select a startingpoint as near as possible to one corner of the circumscribed rectangle.

On putting in wheels corresponding to a ratio which is approximately that of two small integers, the curves will gradually change of themselves, and will be found to cover with shading the whole surface of a certain rectangle. The commencement of this process is exhibited in the last figure of Plate III., the ratio here being approximately that of equality.

If, instead of the paper being fixed to the frame, it is slowly drawn on by the roller, the curves are somewhat distorted, but the order of succession is clearly put in evidence, and the working is much more rapid. The traces thus obtained, five specimens of which are given in Plate III., often bear a striking resemblance to letters of ordinary writing, and might be taken as the foundation of a natural alphabet of quickly-written characters. The approximate ratios are indicated on the left hand of the Plate, and the rigorous ratios on the right, the number of vertical vibrations being in each case greater than the number of horizontal. The horizontal amplitudes are equal to the vertical amplitudes. All the curves except the first are on the same scale, both as regards amplitude and the action of the roller in drawing the paper onwards. In the first curve, the amplitudes are much larger in comparison with the motion

due to the roller, and hence the intersections are more

numerous.

95. A more elaborate combination of parallel simple harmonic motions is furnished by the tide-predicting machine of Sir William Thomson.

The variation of tidal level at a given port is approximately the resultant of two simple harmonic variations, their periods being respectively half a lunar day and half a solar day, and the amplitude of the former being in general rather more than double that of the latter. When the phases of the two concur we have spring tides, and when they are in opposition we have neap tides.

To obtain a closer approximation, the variation of tidal level must be regarded as the resultant of a much larger number of simple harmonic components, the periods of which are known from astronomical considerations, and are the same for all ports, while the amplitudes and the epochs of maximum for the separate components will be very different for different ports. These epochs and amplitudes for a given port can be calculated from a year's continuous record of tidal level at that port (better from several years' record), and when they have been thus ascertained, the tidal level at any future moment can be predicted from them. The tide-predicting machine is intended for making such prediction in the form of a continuous curve, whose ordinates are the heights of the tide. The principle of its working is illustrated by Fig. 40.