ALONG THE CIRCUMFERENCE OF A CIRCLE. 



451 



a velocity proportional to the whole subtense AB. Putting, for shortness' sake, 

 BC = a, CA = b, we have CP = b sin a, and PB 2 = a 2 — b 2 sin A 2 , that is. 



-Tr oc J (a 2 — b 2 sin A 2 ), and similarly 

 ^ oc J(b 2 - a 2 sin B 2 ) . 



When a body projected from N, with the 

 velocity due to a descent through AN, has 

 reached the point a, its velocity there is that 

 which is due to a descent from A to G on 

 the same level with a. Now, if we put A 

 for the diameter NZ, H for the whole height 

 NA, and A for the angle NZa, we have 

 Na = A . sin A, NG = A . sin A 2 , AG = H - 

 A sin A 2 . Wherefore, if g be the intensity of 

 gravitation as measured by the velocity which 

 a falling body acquires in one second, 



V n = 



A.^A 

 dt 



= J'2g. V(H-A.sinA 2 ), or 



dA 



J(2g) • dt 



= n/(5 - A" Sln A 





A 









Fig. 1. 



Z 











B 







/ K 







Xf 





fc<a 





Q. 









H 





\J>' 









■^v 1 * 





Hence, if we assume in our trigon ABC, 



^A 



dt 



= PB^y, 



dA 



the expression -7- = J'2g . J [a 2 — b 2 sin A 2 ) becomes identic with 



dt 



<LA 



dt 



/ XT 1 X 



= J '2g J i ^ — "a sin A 2 Jon puttin 



l 3 



& 2 



H = 



a 2 . a 2 



ir- 



In order, then, to obtain the mechanical arrangement typified by the 

 variable trigon ABC, we must describe a circle having its diameter inversely 

 proportional to the square of AC, and suppose the initial velocity to be that 

 due to a descent through a height exceeding this diameter in the ratio a 2 : b 2 . 

 This gives us the motion represented by the variable angle A. Again, we make 

 another circle having its diameter inversely proportional to a 2 , and take a height 

 less than this in the ratio of b 2 :a 2 ; the oscillatory motion in this arc is repre- 

 sented by the variations of the smaller angle B ; and the time of describing 



