VOL. LXX.] PHILOSOPHICAL TKANSAC TIONS. 729 



tinue to move in a right line, as that point would not be disturbed by the rotatory 

 motion : yet, as in the case we are now considering, the bodies begin to revolve 

 about a dilFerent centre, it may be proper to examine more accurately into this 

 matter, and to show, from -what principle it is that the motion of the centre of 

 gravity is preserved in a right line. 



Let a motion perpendicular to the rod be communicated to a (fig. 2) and then, 

 by cor. 3, prop. ], b will not be disturbed by such an action ; and a will in the 

 first instant have a tendency to revolve about e as a centre, and would actually 

 describe the arc ah, if the body b were fixed : let the angle abh be supposed 

 infinitely small ; and let gk be the arc the centre of gravity would have des- 

 cribed ; and draw the tangents af, Gg to the arcs ah, gk respectively. Now, 

 if A could have moved freely, it would (because af ^ ah) have described af in 

 the same time the arc ah was described, on supposition that b was fixed ; for the 

 radius ba being perpendicular to the circular arc ah, the force of the lever could 

 have no efficacy to accelerate or retard the motion of a in the arc ah, and there- 

 fore the velocity in that arc is the same as it would have been if it had moved 

 freely in the tangent : hence hp is that space through which the centrifugal 

 force of A would have carried that body, could it have moved freely ; but as a is 

 connected to b by means of the lever, it is manifest that the same force which 

 would have carried a from h to f in the direction of the lever, will, when it has 

 both bodies to move, carry it over a space which is to fh, as a : a -f- b, or as 

 Eg : bh, or as gK : fh ; hence that space, or the space through which the cen- 

 trifugal force of A will draw the lever in the direction bh, is equal to Kg ; that 

 is, the point k, which is the centre of gravity of a and b, will be found at o-, 

 and consequently the centre of gravity has preserved its motion uniform in the 

 right line g^, inasmuch as the centrifugal force, acting perpendicularly to the 

 direction of the centre of gravity, can neither accelerate nor retard its motion. 

 In the same manner it may be proved, that the motion of the centre of gravity 

 is continued uniform in the same right line, whatever be the position of the 

 lever. Also, as the centrifugal force acts in the direction of the lever, it cannot 

 alter its angular velocity, which will therefore remain as in ipso motus initio. If 

 now we suppose that, to the force impressed on a, two other equal accelerative 

 forces be communicated to a and b at the same time, it is evident that no altera- 

 tion can arise from the actions of the bodies on each other ; and the case will 

 then be similar to the motion of the bodies, supposing a single force had been 

 impressed at any point d. The like method of reasoning may be extended to 

 any number of bodies. 



The same thing may also be easily demonstrated in the following manner. 

 The centrifugal forces of a and b (fig. 1) are respectively a X ac and b X bc ; 

 also the centrifugal force of the point g, considering it as having both bodies to 



VOL. XIV. 5 A 



