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motion of an infcribed fphere, revolving round the fame fixed axis, 

 and veith the fame angular velocity, zs\ p to r\p^, or as i6 to ip, 

 that is, as four equal fquares to three circles infcribed in them. 



Let the quantity of matter in an indefinitely flender ring, fur- 

 rounding the fphere and cylinder at their common contad A O C, 

 be reprefented by the letter m, its velocity vsrill be as A O = i } and 

 its motion = m, and therefore the motion of the cylinder is to the 

 motion of the ring as j/ to m, or as 2/ to 3 



; m. 



The motion of the annulus, uniformly continued round the axis 

 of the cylinder, is to its motion revolving uniformly in the fame pe- 

 riodic time round one of its diameters, as the circumference of a 

 circle to twice the diameter. 



For (fig. 2) let A R = z, and let its fluxion z be given, R B 

 =: J', A B = X, and A O = r ; let the motion be performed round the 

 diameter A C, the velocity of the point R will be as R B or ^ ; there- 

 fore the fluxion of themotion of the annulus round the diameter AC, 

 is to the fluxion of the motion round the center O in an immoveable 

 plane, a.s z y to s r, that is, from the nature of a circle, as a; to z j 

 and therefore the motions themfelves are to each other in the fame 

 ratio, that is, when :v = A C, as the diameter to half the cir- 

 cumference, or as twice the diameter to the circumference of a 

 circle. 



Hence, by compounding all thefe ratios, the truth of the 

 Lemma is manifeft. 



But 



