196 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1914. 
Now I would direct attention to these rods with arrowheads, 
which are screwed to the gyrostat case. ‘This curved one shows the 
direction in which the flywheel is spinning. The straight rods are 
intended to represent the spin momentum and the tilting action, 
respectively. Both are completely known when their amounts and 
their planes are known. The spin momentum is got by multiplying 
two numbers together, one representing the spin inertia of the wheel 
(which is greater the more the mass is placed in the rim), the other 
the speed of turning. The turning action or ‘‘couple” is also got by 
multiplying the force with which I push by the arm or leverage of the 
force about the axis. So then we represent these two by lines drawn 
at right angles to the two planes, making the lines of lengths to rep- 
resent the two products. Standing on one side of the plane of the 
flywheel, you see it turning against the hands of a clock; standing 
on one side of the plane of the turning action which I apply, you 
observe that action tending to turn the body also against the hands 
of a clock. The two lines representing the two products drawn 
toward you from the two planes represent also the directions of the 
turning actions of the couples. For example, the direction of rota- 
tion of the flywheel being that shown by the curved rod, the line 
representing the spin momentum points outward from the side of 
the gyrostat to which the rods are attached. I call this the spin axis. 
The other line, representing the turning action which I applied, I eall 
the couple axis. 
Now, observe that I set the couple axis so as to point toward your 
left. I push down the side of the gyrostat nearest me, and you see 
that the spin axis turns toward the left. Again, [ turn the couple 
axis so as to point to your right. When so placed it represents a 
turning action tending to depress the end of the axle of the flywheel 
that is nearer you. I apply such an action, and the spin axis turns 
toward your right. In both cases the spin axis turned toward the 
instantaneous position of the couple axis. 
Now I set the couple axis vertical, pointing up. It represents a 
turning action tending to produce horizontal turning in the counter- 
clock direction as seen from above. I apply such an action to the 
fork, when you see that the gyrostat turns the spin axis toward the 
upward direction. Finally, I set the couple axis vertical but point- 
ing down, as in figure 1. It now represents a turning action tending 
to produce clockwise rotation as viewed from above, counterclock 
rotation as seen from below. I apply the action represented and the 
gyrostat turns the spin axis toward the downward direction. 
These experiments may be summed up as follows: The flywheel is 
spinning about axis 1. Any attempt to tilt the gyrostat about axis 
2 produces turning about 3; an attempt to tilt it about 3 produces 
turning about 2. This response of the body seems paradoxical, but 
