Wakelin. — Fallacies in the Theory of Circular Motion. 139 



These two equations may be put in the form of a ratio, as follows : — 

 AsAP : PB : : tv : ^ff^ 



or in a better form : 



AP _ tv (4) 



PB - i/t2 

 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 : — 



_(AP)2__M2 (5) 



PB - J/t2 



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 : 



2?--PB _ t'^v^ (6) 

 PB i/t2 



or, as PB cancels out, the simple form will be : 



It will be necessary to pause here. A careful study of these three last 

 eqiiations, namely (5) (6) and (7), shows us that t^v^ is the square of the 

 tangent, and that ^fl^ 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 2r is, of course, the diameter of the circle. In the last three 

 equations the quantity t^ 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 -J 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 : 



^t^ (8) 



It may be read thus : If the value of the square of the tangent be divided 



by the value of tivice the distance fallen through, the quotient will give 



the value of the radius of the circle. Cancelling out the time equation 



(8) becomes 



v^ (9) 



and from this equation the formula for the measure of centripetal force is 

 obtained, that is the value of / is found to be as follows ; 



/=:!^ (10) 



