Waxetin.— Fallacies in the Theory of Circular Motion. 139 
These two equations may be put in the form of a ratio, as follows :— 
Ma A&P: PR.:i wee 
or in a better form: 
AP iv (4) 
PB = a7 
PB 
Both sides of this equation are really identical, the capital letters forming 
one side of the equation being lines, and the small letters forming the other 
side of the equation being the algebraical,—that is the numerical,—value of 
those lines. The square of the tangent AP is equal to the square of tv. 
See equation (2). Equation (4) can now be put as follows :— 
Ba Are a eg) 
PB ~ hfe 
On referring back to equation (1), it will be seen that (AP)? is equal to 
2r°PB. Substituting this value of (AP)? in equation (5), we have now: 
‘PB _t?v2 (6) 
PE 30 
or, as PB cancels out, the simple form will be : 
Qr = _t%0? (7) 
3 ft? 
It will be necessary to pause here. A careful study of these three last 
equations, namely (5) (6) and (7), shows us that év? is the square of the 
tangent, and that 4 f@ is the distance fallen through. The last equation then 
reads thus, if the numerical value of the square of the tangent be divided by 
the numerical value of the distance fallen through, the quotient will be equal 
to 2r. Here 2 is, of course, the diameter of the circle. In the last three 
equations the quantity @ could have been cancelled out, but, by retaining 
this quantity, the whole algebraical expression on the right-hand side of the 
equation can be directly transformed into its geometrical equivalent. The 
term 4 ft? is certainly the distance fallen through represented by the line 
PP, and is it not equally true that tv is the length of the tangent repre- 
sented by the line AP, t?v? being the value of the square on the tangent, 
Which is equal to (AP)?. The fraction in the denominator of the fraction on 
the right-hand side of equation (7) is got rid of, and the equation will then 
Stand thus: 
t2y2 (8) 
il t 
It may be read thus: If the value of the square of the tangent be divided 
by the value of twice the distance fallen through, the quotient will give 
the value of the radius of the circle. Cancelling out the time equation 
8) becomes 
debi 
and from this equation the formula for the measure of centripetal force is 
» that is the value of / is found to be as follows; 
(10) 
y2 
i= 
