Page:CarmichealMass.djvu/6

He takes a rod of two units length, whose mass is so small as to be negligible, and attaches to its ends two balls of equal mass. Then he suspends this rod by a wire so as to form a torsion pendulum. We assume that the line of relative motion of the two systems is perpendicular to the line of this wire.

Let us consider the period of this torsion pendulum in the two cases when the rod is clamped to the wire so as to be in equilibrium in each of the following two positions: (1) With its length perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$; (2) with its length parallel to this line of motion.

As B observes it the period must be the same in the two cases; for, otherwise, he would have a means of detecting his motion by observations made on his system alone, contrary to postulate M. Then from the relation of time units on $S_{1}$ and $S_{2}$ it follows that the two periods will also appear the same to A. As observed by B the apparent mass of the balls is the same in both cases. We inquire as to how they appear to A. Let $m_{1}$ and $m_{2}$ be the apparent masses, as observed by A, in the first and second cases respectively. It is obvious that $m_{1}$ is the longitudinal mass and $m_{2}$ the transverse mass of the balls in question.

When the pendulum is in motion it appears to B that each ball traces a circular arc. From the relations between units of length in the two systems it follows that to A it appears that the balls trace arcs of an ellipse whose semi-axes are 1 and ${\sqrt {1-\beta ^{2}}}$ and lie perpendicular and parallel, respectively, to the line of relative motion of the two systems.

Let us now determine the period of each of these two pendulums as they are observed by A. By equating the expressions for these periods we shall find the relation which exists between $m_{1}$ and $m_{2}$.

Let x and y be the cartesian coordinates of a point as determined by A, the axes of reference being the major and minor axes of the ellipse in which the balls move. Let $x'$ and $y'$ be the coordinates of the same point as determined by B. Then the circular path of motion, as determined by B, has the equations

$x'=\cos \theta ,\ y'=\sin \theta$

the angle θ being measured from the major axis of the ellipse. The equations of the ellipse, as determined by A, are

$x=\cos \theta ,\ y={\sqrt {1-\beta ^{2}}}\sin \theta$

In the first case — when the rod is perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$ — the amount of twisting in the wire when the ball is in a given position is the absolute value of the corresponding angle θ;