VOL. XXXVII.] PHILOSOPHICAL TRANSACTIONS. 483 



appear by moving the balance into the position ab; which shows the velocity 

 of p to be the perpendicular line ea, and the velocity of b will be the perpen- 

 dicular line bg: for if the weights p and w are equal, and also the lines eaand 

 bg, their momenta made up of ea multiplied into w, and bg multiplied into p, 

 will be equal, as will appear, by their destroying each other in making an 

 equilibrium. But if the body w was removed to m, and suspended at the point 



D, then its velocity being only fd, it would be over-balanced by the body p; 

 because fd multiplied into m, would produce a less momentum than p multi- 

 plied into bg. 



As the arcs Aa, Bb, and od, described by the ends of the balance, or points 

 of suspension, are proportionable to their sines ea, gb, and df, as also the radii 

 or distances ca, cb, and cd; in the case of this common sort of balance, the 

 arcs described by the weights, or their points of suspension, or the distances 

 from the centre, may be taken for the velocities of the weights hanging at a, 

 B, or D; and therefore the acting force of the weights will be reciprocally as 

 their distances from the centre. 



Scholium. — The distances from the centre are taken here for the velocities 

 of the bodies, only because they are proportional to the lines' ea, bg, and fd, 

 which are the true velocities. For there are a great many cases in which the 

 velocities are neither proportionable to the distances from the centre of motion 

 of a machine, nor to the arcs described by the weights or their points of sus- 

 pension. Therefore it is not a general rule, that weights act in proportion to 

 their distances from the centre of motion ; but a corollary of the general rule, 

 that weights act in proportion to their true velocities, which is only true in 

 some cases. Therefore we must not take this case as a principle, which most 

 workmen do, and all those people who attempt to find the perpetual motion, 

 as is most amply shown in the Philos. Trans. N° 36g. 



But to make this evident even in the balance, we need only take notice of 

 the following experiment, fig. 3 ; where acbkkd is a balance, in the form of 

 a parallelogram, passing through a slit in the upright piece no, standing on the 

 pedestal m, so as to be moveable on the centre pins c and k. To the upright 

 pieces ad and be of this balance are fixed, at right angles, the horizontal pieces 

 FG and hi. That the equal weights p, w, must keep each other in equilibrio, 

 is evident; but it does not at first appear so plainly, that if w be removed to v, 

 being suspended at 6, yet it shall still keep p in equilibrio; though the experi- 

 ment shows it. Nay, if w be successively moved to any of the points ], 2, 3, 



E, 4, 5, or 6, the equilibrium will be continued; or if, w hanging at any of 

 those points, p be successively moved to d, or any of the points of suspension 

 on the cross piece fg, p will at anv of those places make an equilibrium with 



3 a 2 



