8 PHILOSOPHICAL TRANSACTIONS. [anNO 1713. 



particle p, situated in the point z: for from these forces jointly, arises the 

 absolute motion of the whole plane. By means of this, is given the motion of 

 every point proposed ; whence in its turn is found that point whose motion 

 is given. 



But the particle p will be urged by the force of its own gravity, which, if 

 the cohesion of the particles were dissolved, in a given very small time, would 

 produce a given acceleration of motion, in the perpendicular to the horizon 

 zy. Draw ocy perpendicular to ez, and the acceleration zy will be resolved 

 into the parts zx and ocy. Because of the rigidity of the body, the force zx 

 will be taken away by the resistance of the point c. But by the remaining 

 force xy the space abd is turned about the point c. And drawing the horizontal 

 line CO, and a perpendicular z*, it will be as — : viz. because of the given 

 force of gravity, and the similar triangles xyz and scz. Therefore the force of 

 the particle /), to move the space abd, will be as — X p. 



To collect these forces together, let o be an invariable point, in a line drawn 

 at pleasure, and at a distance co as yet unknown. Then the force of the par- 

 tide p to move the point o, will be as -^ x — X /J, that is, as — X p. And 



CO C2 



CO . . C« 



the acceleration which p contributes to the same point o, will be as — x — . 



Therefore the force — X p being applied to the acceleration ^—, the quo- 



tient will be — ^ X p, which, if it be supposed to move in the point o with the 



same acceleration — , would produce just the same motion, as the particle 



p produces in the same point o. Thus finally the problem is reduced to a well- 

 known theorem of motion : for the sum of the forces — X p being applied to 



the sum of the particles — -, X p, the quotient will give the absolute accelera- 

 tion of the point o. Then drawing the perpendicular oo, and supposing this 

 acceleration to be equal to the given acceleration — of the point o, the distance 



CO will be given. For let — =z d, and by the method of fluxions it is 

 C5 X /> = M, and cz^ X p = c. Then because of co being Variable, the sum of 

 all the forces will be — X P = — , and the sum of all theparticles — •, X i& = — . 



CO •' CO '^ CO* ^ CO* 



Hence, applying the sum of the moments to the sum of the bodies, it will 

 be - X cof/, and therefore co = — . Therefore, c and m being found, co 

 will be given by the inverse method of fluxions, q.e. i. 



Corol. From the centre of gravity g draw g^ perpendicular to the horizontal 



