Proof of the Principle of Virtual Velocities, 17 



not sufficiently explained. The following attempt to meet the 

 difficulty complained of, is offered to the consideration of those 

 who may wish to see a proof, in other respects remarkable for 

 elegance and brevity, free from every objection. 



Lagrange's reasoning is of the following nature. Instead 

 of the given forces he substitutes other equivalent forces, in a 

 manner equally applicable to all cases of equilibrium. The 

 way in which he does this, though not the only one that might 

 be adopted, is perhaps the best. A system consisting of two 

 blocks of pulleys is placed so that one block is attached to the 

 point of application of one of the forces, as P, and the other 

 to an arbitrary point taken in the line in which P acts. A 

 cord, having a weight w attached to one extremity, passes 

 over a fixed pulley, that the weight may hang vertically, and 

 is then carried over the pulleys of the blocks, forming m strings 

 between them. The continuation of the same cord is then 

 made to pass over the pulleys of another system, situated with 

 respect to another force Q as the first was with respect to P ; 

 and the strings between the blocks of this system are m! in 

 number. The same thing is done with respect to all the forces, 

 and the other extremity of the cord is attached to a fixed 

 point. That the strings between the blocks may be parallel, 

 we may conceive the blocks and pulleys to be indefinitely small. 

 The tension of the cord will be the same throughout, and 

 equal to w. Hence if m w = P, n n' w = Q, m n w = R, &c, 

 the effect of the systems of pulleys will be exactly the same 

 as that of the given forces. This supposes the forces to be 

 commensurable one with another : if they are not so, we may 

 take w as small as we please, and so make the substituted 

 forces as nearly equal to the given forces as we please. 



The substitution being thus made, Lagrange goes on to 

 say : — 4< It is evident that in order that the system drawn by 

 these different (substituted) forces may remain in equilibrium, 

 it is necessary that the weight (to) should not be able to descend 

 by any infinitely small displacement whatever of the points of 

 the system; for the weight tending always to descend, if there 

 be a displacement of the system which permits it to descend, it- 

 will descend necessarily and produce this displacement. 11 This 

 is the part of the proof to which I have alluded above; and 

 certainly the reason here given for the immobility of w is not 

 easy of comprehension. 



The reason that w neither ascends nor descends may I 

 think be seen, if the following principle, which may be consi- 

 dered a definition of equilibrium, be admitted : When a rigid 

 mass is held in equilibrium by any forces, it may receive any 

 indefinitely small displacement whatever, when it is not re- 



Third Series. Vol. £ 2. No. 7. Jan. 1833. ' D 



