The versed sine of   1” is 0.000,000,000,011,752
"   "      "   "    2” is 0.000,000,000,047,008
"   "      "   "    3” is 0.000,000,000,105,768
"   "      "   "    4” is 0.000,000,000,188,032
"   "      "   "    5” is 0.000,000,000,293,805
"   "      "   "    6” is 0.000,000,000,423,072
"   "      "   "    7” is 0.000,000,000,575,848
"   "      "   "    8” is 0.000,000,000,752,128
"   "      "   "    9” is 0.000,000,000,951,912
"   "      "   "   10” is 0.000,000,001,175,222
"   "      "   "  100” is 0.000,000,117,522,250

You observe the deflection for 10” of arc is 100 times as great, and for 100” of arc is 10,000 times as great as it is for 1” of arc.  So far as is shown by the 15th place of decimals, the versed sine varies as the square of the angle; or, in a given circle, the deflection, and so the centrifugal force, of a revolving body varies as the square of the speed.

The reason for the third law is equally apparent on inspection of Fig. 2.  It is obvious, that in the case of bodies making the same number of revolutions in different circles, the deflection must vary directly as the diameter of the circle, because for any given angle the versed sine varies directly as the radius.  Thus radius O A’ is twice radius O A, and so the versed sine of the arc A’ B’ is twice the versed sine of the arc A B. Here, while the angular velocity is the same, the actual velocity is doubled by increase in the diameter of the circle, and so the deflection is doubled.  This exhibits the general law, that with a given angular velocity the centrifugal force varies directly as the radius or diameter of the circle.

We come now to the reason for the fourth law, that, with a given actual velocity, the centrifugal force varies inversely as the diameter of the circle.  If any of you ever revolved a weight at the end of a cord with some velocity, and let the cord wind up, suppose around your hand, without doing anything to accelerate the motion, then, while the circle of revolution was growing smaller, the actual velocity continuing nearly uniform, you have felt the continually increasing stress, and have observed the increasing angular velocity, the two obviously increasing in the same ratio.  That is the operation or action which the fourth law of centrifugal force expresses.  An examination of this same figure (Fig. 2) will show you at once the reason for it in the increasing deflection which the body suffers, as its circle of revolution is contracted.  If we take the velocity A’ B’, double the velocity A B, and transfer it to the smaller circle, we have the velocity A C. But the deflection has been increasing as we have reduced the circle, and now with one half the radius it is twice as great.  It has increased in the same ratio in which the angular velocity has increased.  Thus we see the simple and necessary nature of these laws.  They merely express the different rates of deflection of a revolving body in these different cases.