THE EFFECTS OF MACHINES WHEN IN MOTION. 145 



act. Hence it follows in both cafes, that if the y 

 of the same beam or body, be reduced to different 

 diftances, its value will be im'erseli/ as the squares 

 of these distances. 



Prob. 8. Let A be the area of the circle whofe 

 diameter is unity, and r = the radius of the circular 

 plane ABC: and let j& represent the periphery of 

 a circle, or a ring into which we will conceive as 

 many particles collected, as, with any angular ve- 

 locity, shall have the same force, as the mass of the 

 circular plane, (or solid wheel of the same diameter, 

 and uniformly thick, ) collected into a circle, whose 

 radius is the distance of the center of gyration from 

 the center C, moving with the same angular ve^ 

 locity : the value of p' is required. 



Now it is evident from the 

 nature of the problem, that 

 pr~ will be equal to all iht pd^ 

 in ABC. And since 4 A r'^ is 

 the area of ABC, we have 



w'' zig,andj&z:re/"X4A=eA7- 



by substituting the value of iv' 

 which value is equal half the 

 mass of A B C, whether it be a 

 circular plane or solid wheel. 



Now this power p' may be either a ring, as is here 

 conceived, or a weight equal to that of the ring, di- 

 vided into two equal parts, each acting at the ex- 

 tremity of a lever, revolving on its center, and 

 whose length is equal to the diameter of the ring ; 

 and in the same manner we may conceive the p' in 

 problem 7 to be resolved into a ring of equal weight 

 whose diameter is ecjual A B. 



Vol. VI. K Prob. p. 



