138 Transactions. — Miscellaneous. 



{Euclid, Third Book, Proj). 36.) The particular case taken is where the 

 point outside the circle is the extremity of the tangent, and a diameter pro- 

 duced to meet the point. Mr. Todhunter deduces the formula apparently 

 from the parallelogram of velocities. 



The method of deriving the formula for the measure of centripetal force 

 is very clearly and precisely given by Mr. (I think now Professor) Goodeve 

 in his " Principles of Mechanics." He takes the thirty-sixth Proposition of 

 the Thu'd Book of Euclid, where it is proved that the square of the tangent 

 is equal to the rectangle contained by the diameter produced to meet the 

 tangent and the part produced. Accordingly, in the annexed figure, the 

 square on the line AP will be equal to the rectangle DP, PB. In the 

 work referred to, the explanation given is nearly as 

 follows : — The rectangle DP, PB is equal to DB, 

 PB together with the square on PB. When the 

 angle is made very small, the square on PB may be 

 neglected, and then the square on the tangent AP 

 is equal to the rectangle DB, BP. Now DB is the 

 diameter of the circle; then (AP)^:=2r • PB. (1.) 

 In the limit this is mathematically exact. Let the 

 body revolving with uniform motion be supposed passing through the point 

 B by the end of the time t. If no force had deflected the body it would have 

 pursued a straight course along the tangent, and would have reached the 

 point P at the end of the time t, or to speak more exactly it would very 

 nearly have reached that point, because AP the tangent is greater than 

 AB the arc. When the arc, however, is extremely small, the difference 

 between the arc of the angle and its tangent is inappreciable — in the limit 

 they coincide. The body was deflected from its course the length PB. It 

 is pulled through the distance PB, that is it falls through that distance. 

 From this geometrical construction we can now derive an algebraical equa- 

 tion. The liiie AP is equal to tv, and the distance fallen through, namely 

 PB, is equal to -J- ff. Here v of course is the velocity of the body, and / 

 stands as usual for the acceleration. The time that would have been taken 

 by the body to move from A to P is, of course, the same that it took to fall 

 from P to B, that is, if the angle represented by the arc, or tangent, be very 

 small. In the figure the angle is very much exaggerated for the sake of 

 clearness, but the arc taken should not be greater than a degree when the 

 (error will be very small. Bearing these considerations in mind, we can 

 proceed to evolve fi-om the equation we have obtained the measure of centri- 

 petal force. The equation before given may be conveniently put distinctly. 



Thus :— 



AP = tv (2) 



VB = ift^ (3) 



