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

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

$BC=a\sin {\phi }=a\cos {\left(\phi -{\frac {\pi }{2}}\right)}=acos{\left(\omega t-{\frac {\pi }{2}}\right)},$

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

$x=a\cos {(\phi -\epsilon )}$ and $y=b\cos {\phi }.$

If $\phi$ and $\phi '$ are commensurable, that is, if $\phi '=n\phi$, the curve is re-entrant. Making this supposition,

$x=a\cos {\phi }\,\cos {\epsilon }+a\sin {\phi }\,\sin {\epsilon }$

, 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

$x=y\cos {\epsilon }+(a^{2}-y^{2})^{\frac {1}{2}}\sin {\epsilon },$

from which

$x^{2}-2xy\cos {\epsilon }+y^{2}\cos ^{2}{\epsilon }=a^{2}\sin ^{2}{\epsilon }-y^{2}\sin ^{2}{\epsilon },$

or,

$x^{2}-2xy\cos {\epsilon }+y^{2}=a^{2}\sin ^{2}{\epsilon }.$

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