Page:Elementary Text-book of Physics (Anthony, 1897).djvu/37

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[§ 21
MECHANICS OF MASSES.
23

Returning to our first suppositions, letting be the point from which epoch and time are reckoned, it is plain that, since

the projection of on the diameter also has a simple harmonic motion, differing in epoch from that in the diameter by .

It follows immediately that the composition of two simple harmonic motions at right angles to each other, having the same amplitude and the same period, and differing in epoch by a right angle, will produce a motion in a circle of radius with a constant velocity. More generally, the coordinates of a point moving with two simple harmonic motions at right angles to one another are

and

If and are commensurable, that is, if , the curve is re-entrant. Making this supposition,

, and .

Various values may be assigned to , to , and to . Let equal and equal 1; then

from which

or,

This becomes, when , , the equation for a circle. When , it becomes , the equation for a straight line through the origin, making an angle of with the axis of . With intermediate values of , it is the equation for an ellipse. If we make , we obtain, as special cases of the curve, a parabola and a lemniscate, according as or . If and are unequal, and , we get, in general, an ellipse.

We shall now show, in the simplest case, the result of compounding two simple harmonic motions which differ only in epoch