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In a cylinder, j = i, thei-efore the fluxion of the moment 

 = 1 p x'^ X; whofe fluent is ^ /, when a; = i. 



The moment of a ring revolving round its center is 

 double the momentum of the fame ring revolving round 

 one of its diameters. For let z be the fluxion of the arch, y the 

 ordinate, and x the abfcifla, radius being unity; %y- \% the 

 fluxion of the moment of the ring revolving round one of its 



diameters ; but, from the nature of the circle, z =— , therefore 



y 



zy-:=^xy, which is the fluxion of 'the area ABR; therefore 

 when X— I, that is, when the arch is eqiial to ^^ p, the meafure 

 of the moment will be the area of a quadrant ; and the mea- 

 fure of the moment of the entire ring will be equal to the area 

 of the circle, or ip. 



If the ring revolve round its center, in an immoveable plane, 

 its moment will be equal to the ring multiplied into the fquare 

 of its radius, that is, equal to p Therefore the moment in the 

 former cafe is to that in the latter, as ^^ to^, or as one to two. 



Hence, from what has been demonftrated, the momentum 

 of a fphere is to the momentum of a cylinder, revolving round 

 their axes with the fame angular velocity, as ~^j to i ; the mo- 

 mentum of a cylinder is to the momentum of a ring revolving 

 round its centre, in like manner, as ^ ptom; and the momentum 



Vol. VII. B of 



