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

[§ 21 Returning to our first suppositions, letting $$X$$ be the point from which epoch and time are reckoned, it is plain that, since the projection of $$B$$ on the diameter $$OY$$ also has a simple harmonic motion, differing in epoch from that in the diameter $$OX$$ by $$\frac{\pi}{2}$$.

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 $$a$$ with a constant velocity. More generally, the coordinates of a point moving with two simple harmonic motions at right angles to one another are If $$\phi$$ and $$\phi '$$ are commensurable, that is, if $$\phi ' = n \phi$$, the curve is re-entrant. Making this supposition,, and $$y = b \cos{n \phi}$$.

Various values may be assigned to $$a$$, to $$b$$, and to $$n$$. Let $$a$$ equal $$b$$ and $$n$$ equal 1; then from which  or,  This becomes, when $$\epsilon = 90^\circ$$, $$x^2 + y^2 = a^2$$, the equation for a circle. When $$\epsilon = 0^\circ$$, it becomes $$x - y = 0$$, the equation for a straight line through the origin, making an angle of $$45^\circ$$ with the axis of $$X$$. With intermediate values of $$\epsilon$$, it is the equation for an ellipse. If we make $$n = \tfrac{1}{2}$$, we obtain, as special cases of the curve, a parabola and a lemniscate, according as $$\epsilon = 0^\circ$$ or $$90^\circ$$. If $$a$$ and $$b$$ are unequal, and $$n = 1$$, 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