156 PHILOSOPHICAL TRANSACTIONS. [aNNO 1777» 



acting on the line ocq at the distance g from c, would be equivalent to the 

 force of all the particles in the said section, whose thickness is denoted by the 

 indefinitely small quantity x' ; the distance c^ being denoted by x, and a being 

 put for (.78539) the area of a quadrant of a circle whose radius is 1. 



4. Fig. 22, in the cylinder whose length is 2b and diameter 2r ; y being = r, 

 i/'* — x'^y- will be = r'^ X (i?" — x"^) : consequently the fluent of (J r' — a?') 

 X i', generated while x from o becomes = /', being ^br' — -^h^, we have 



-^- X mny:. {- -) = 1^. X (3r'^ — 4b'^) X M for the force which, acting 



g 4 3 12^' 



as above at the distance g from (c) the centre of gravity of the cylinder, would 

 be equivalent to the efficacy of the forces acting as above on all the particles of 

 the cylinder to turn it about a diameter passing through c, m being the mass or 

 content of the cylinder. 



5. Fig. 23, in the spheroid whose proper axis is 2b and equatorial diameter 



2r, f being = ;:; {l^ -x% ±f - xy will be = ,- X (j - '^\ + % - x^ + 

 -) : consequently the fluent ot — ^- + -^ — x-t + — , generated 



ylj^ ^Ify ,.2^ J3 J) O 



while X from o becomes = b, being — — -^ + 55 — j+j=f3X {r^b — 



b'), we have— ^ X mn X {r'b - b') =-^' X (r' - b') X s for the force which, 

 acting at the distance g from c the centre of the spheroid, would be equivalent 

 to the efficacy of the forces acting as above on all the particles of the spheroid 

 to turn it about a diameter of its equator, s being the mass or content of the 

 spheroid. — These equivalent forces are distinguished by the name of motive 

 forces; the correspondent accelerative forces are computed in the following 

 articles. 



6. Fig. 24, the body being a spheroid whose centre is c, and whose proper 

 axis pn is = 2b, and equatorial diameter ab = 2/- ; let f be the accelerative 

 force of a particle at the distance g from the axis about which the body is urged 

 to turn, which axis is supposed to be a diameter of its equator. Denote c^ by 

 x; ki by^/; and let the abscissa ^o and its correspondent ordinate (parallel to the 

 last mentioned axis) in the circle w hose radius is hi, be denoted by s and t res- 

 pectively. Then, considering the body as urged to turn about that diameter of 

 its equator which is at right angles to ab, the accelerative force of every particle 

 in the said ordinate will be = X f, and the motive force of all the 



particles in the same ordinate will be = ^' ^ ' '- X vt/ = —-„—'- X f/ /(y* 



_ r) ; to which (by the property of the lever) a motive force = -—p— X 

 (,/ ^ (^2 _ gi^^ actuig at the distance g from the centre at right angles to a ray 

 from it, would be equivalent. Therefore, considering a and j/ as invariable, and 



