MECHANICS. 



and C F be called p and w, the condi- 

 tion on which the power P shall support 

 the weight W, will, according to the 

 principle of virtual velocities, be 



P: W: :w:p, 

 or P x p = W x w. 

 The meaning of which is, that the 

 power will sustain the weight in equili- 

 brium, provided that the number of 

 ounces in the power, multiplied by the 

 number of inches in its distance from 

 the centre, shall be equal to the number 

 of ounces in the weight, multiplied by 

 the number of inches in its distance 

 from the centre. It is evident, that any 

 other denominations of weight and 

 measure besides ounces and inches 

 may be used, provided the same denomi- 

 nation be used both with respect to the 

 weight and power. 



(8.) Such is the condition of equili- 

 brium resulting from the principle of 

 virtual velocities, and which it is veiy 

 easy to submit to the test of experiment. 

 Let a weight W, amounting to any 

 number of ounces, be suspended at 

 the point F, and let the number of 

 inches in C F be exactly measured. Sup- 

 pose that 1 2 ounces are suspended, and 

 that C F is 8 inches. Now take any 

 distance C G on the other side, and 

 suppose that distance 32 inches, and 

 that a weight of three ounces be sus- 

 pended, it will be found, that equilibrium 

 shall be preserved, and that the power 

 shall exactly balance the weight. ; and, 

 accordingly, the product of 3 and 32 is 

 exactly equal to the product of 1 2 and 8. 



Again, if instead of 32 inches C G is 

 24 inches, it will then be found to re- 

 quire a power of 4 ounces to balance 

 the same weight. The product of 4 and 

 24 is 96, as before. In the same way, 

 however we may change the distance of 

 the power from the centre, it will be 

 necessary to change its amount, so that 

 the product of the number of ounces in 

 it, by the number of inches in the dis- 

 tance, shall be equal to 96, in order 

 that it shall exactly balance the weight. 

 If in any case the product exceed 96, 

 the power will preponderate ; and if the 

 product be less than 96, the weight will 

 preponderate. 



It appears, therefore, that the same 

 weight W, at the same distance C F 

 from the centre, may be balanced by 

 innumerable different powers. In fact, 

 a power of any magnitude whatever, 

 great or small, may balance it, provided 

 that the distance of that power from the 

 centre be so regulated, that when mul- 



tiplied by the power itself, the product 

 shall be equal to the product of the 

 weight multiplied by its distance from 

 the centre. 



It is evident, that the efforts which 

 the power and weight make to turn 

 the machine round the centre C, will 

 be the same, to whatever point in the 

 lines G w or Fm', the strings sup- 

 porting the power and weight may 

 be attached, or even though they be 

 attached to points in the lines G n 

 and ~Fn' above the points G and F. 

 Thus it appears, that in estimating the 

 distances of the power and weight from 

 the centre, we are not to take the dis- 

 tances of the points of suspension ; but 

 the perpendiculars drawn from the cen- 

 tre C to direction n m and n' in' in which 

 the power and weight act, Thus, if 

 the power and weight were suspended 

 from n and m', we should still consider 

 C G and CF to be their distances from 

 the centre. 



In like manner, the directions of the 

 power and weight may not happen to be 

 parallel, as in the example we have 

 taken ; but still their distances from the 

 centre of motion are estimated by per- 

 pendiculars from that point upon their 

 directions. 



Fig. 2. 



Let the point of application of the 

 weight be L {fig. 2.) and let the string by 

 which the weight acts pass over a wheel 

 H ; and in like manner let the power act 

 by a string at K passing over a wheel I. 

 In this case, L H is the direction of the 

 weight, and K I that of the power. Sup- 

 pose the perpendicular C F and C G 

 drawn upon their directions ; the condi- 

 tion of equilibrium will still be the same ; 

 viz that the product of the power P, 

 and the perpendicular C G shall be 

 equal to the product of the weight W 

 and the perpendicular C F. This may 

 easily be established experimentally. 



