VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 731 



Prop. 6. To determine the time of one revolution, supposing every thing given 

 as in prop. 3. — The point D being given, we have, from cor. 2 to the last prop. 



CG = . Put w equal the circumference of a circle whose radius is cg; 



then it appears from the last prop., that w is the space the centre of gravity 

 passes over in the time of one revolution; hence, because from prop. 4, the 

 centre of gravity moves uniformly, we have, by prop. 3, 



2XVXPXCG ,„ ,, 2XVXPXCG 4.1 i- f 



l" :: w : w X , — ; — ^ = the time of one re- 



(a -h b) X CG + P X DC ■ (A + b) X CG + P X DC 



volution. 



Cor. Hence the angular velocity, being inversely as the time of a revolution, 



.,, (a 4- P,) X CG + P X DO 



will vary as . 



■' V X P X CG X W 



Prop. 7- The point c, as the centre of gravity moves forward, will describe 

 the common cycloid. — Fiom the description of the common cycloid, it appears 

 that the centre of the generating circle passes over a space equal to the circum- 

 ference of that circle while it makes one revolution. With the centre g (fig. 3) 

 and radius oc, describe the circle cxy, and draw cr, gw, perpendicular to abc, 

 and let the circle cxy be supposed to revolve on the line or: then will the centre 

 G move over a space equal to the circumference of the circle cxy while it makes 

 one revolution, and the point c will describe the common cycloid; but, from 

 prop. 5, the point g will move over a space equal to the circumference of a circle 

 whose radius is gc, while the bodies, and consequently go, make one revolu- 

 tion; and hence the point c will describe the same curve as before, that is, the 

 common cycloid. 



Prop. 8. Let a motion be communicated to the lever obliquely; to determine 

 the point about which the bodies begin to revolve. — Let fd (fig. 4) represent the 

 force communicating the motion at the point d, which resolve into 2 others, fh, 

 HD, the former fh parallel to the lever, and the latter hd perpendicular to it. 

 Let c be the point about which the bodies would have begun to revolve, had 

 the force hd only acted, and which may be found by prop. 1 : and suppose in this 

 case mgn to have been the next position of the lever after the commencement 

 of the motion, or that the bodies a, b, and centre of gravity g, had been car- 

 ried to m, g, and n respectively. But as the force fh acts at the point d at the 

 same time in the direction of the rod, if we take g^ : eg as fh : hd, then while 

 the centre of gravity would have moved from g to ^ in consequence of the force 

 HD, it will by means of the force fh be carried in the direction of the lever from 

 G to q, and also every other point of the lever will be carried in the same direc- 

 tion with the same velocity; take therefore xp and Br each equal to g^, and com- 

 plete the parallelograms A.a, gw, and Bb; then the bodies a, b, and centre of 

 gravity g will, at the end of that time, be found at a, b, and w respectively, 

 and atvb will be the position of the lever. Now it is evident that c is not the 



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