[ 6 ] 



By the revolution of the circle E A H C, and cii-cumfcribed 

 fquare (fig. 2.) P Q S T round the common axis E H, lee 

 there be delcribed a fphere and circumfcribed cylinder. Let 

 the radius A O be == t, the periphery of the circle A E CH 

 = /, the ordinate B R =y, abfciffa B O = .v. Then \ : p:: x: pXy 

 the periphery of the circle whofe radius is O B ; therefore 

 f> X X 2y will be the furface generated by the ordinate R G, 

 in the revolution of the circle A E C H round the dia- 

 meter E H : but X will be the meafure of the velocity of the point 

 B, therefore 2. p x- y will be the momentum of all the particles 

 in that furface; and the fluent of the quantity 2 p x'^ y x will 

 be the niomentum of the entire fphere, when x is equal to 

 the radius A O. But y= i — x'}^ ; therefore the fluxion 



X* xy = X' XX I — x^'= ■ ■ — .1,== ; and the fluent 



X"" X 



Y3^i — I X circular arc ER — ^ x x i — a'I', and the 

 fluent of 



-- ^■.■-= — 3 X circular arc E R — 2 x = + ■^ x xx i —x ^^ ; 



X-,^ _ : 



therefore the whole fluent, when x = i, is | x quadrantal arc 



E A — ~p; and 2p x- x x i — x-,i = J_ p-, the motion of 

 the entire fphere. 



In a cylinder, the ordinate y becomes = B R = i ; therefore the 

 fluxion of the momentum of the cylinder = 2p x' x, whofe fluent, 

 when .V = r, is fp. Therefore the motion of a cylinder is to the 



motion 



