112 



Sir W. Thomson on a Machine for the [Dec. 5, 



. . . (21), &c, do not vary sensibly from the values which they have 

 Avhere x u x 2 , . . . x n , are each infinitely small. In practice it will be 

 convenient to so place the axes of Bi, B 2 , . . . B, 2 , and the mountings 

 of the pulleys on Bi, B 2 , . . . B M , and the fixed points Di, Bi, D 2 , &c, 

 that when x x , x 2 , . . . x n are infinitely small, the straight parts of each 

 cord and the lines of infinitesimal motion of the centres of the pulleys 

 round which it passes are all parallel. Then J(ll), i(21), . . . \{n) 

 will be simply equal to the distances of the centres of the pulleys 

 Pii, P215 • • • P»j from the axis of Bi ; 



|(12), i(22) . . . \(n$) the distances of P l2 , P 22 , . . . P M2 from 

 the axis of B 2 , and so on. 



In practice the mounting of the pulleys are to be adjustable by 

 proper geometrical slides, to allow any prescribed positive or negative 

 value to be given to each of the quantities (11), (12), . . . (21), &c. 



Suppose this to be done, and each of the bodies B x , B 2 , . . . B» to 

 be placed in its zero position and held there. Attach now the cords 

 firmly to the fixed points D : , D 2 , . . . D» respectively ; and passing 

 them round their proper pulleysj bring them to the other fixed points 

 E 1? E 2 , . . . E», and pass them through infinitely small smooth rings 

 fixed at these points. Now hold the bodies B 1? B 2 , . . . each fixed, 

 and (in practice by weights hung on their ends, outside E x , E 2 , . . . 

 E M ) pull the cords through Ei, E 2 , . . . E» with any given tensions* 

 Ti, T 2 , . . . T„. Let Gi, G 2 , . . . G M be moments round the fixed 

 axes of Bi, B 2 , . . . B n of the forces required to hold the bodies fixed 

 when acted on by the cords thus stretched. The principle of "virtual 

 velocities," just as it came from Lagrange (or the principle of "work"), 

 gives immediately, in virtue of (I), 



G- 1 =(11)T 1 + (21)T 3 + . . . + (nl)T n 

 G 2 =(12)T 1 '+(22)T 3 + . . . +(n2)% 



. G»=(in)T 1 +(2*)T,+ . .... + WT» 



Apply and keep applied to each of the bodies, B x , B 3 . . . B» (in 

 practice by the weights of the pulleys, and by counter-pulling springs), 

 such forces as shall have for their moments the values Gi, G 3 . . . G M , 

 calculated from equations (II) with whatever values seem desirable for 

 the tensions Ti, T 2 . . . T«. (In practice, the straight parts of the 

 cords are to be approximately vertical, and the bodies B x , B 2 , are to 

 be each balanced on its axis when the pulleys belonging to it are 



* The idea of force here first introduced is not essential, indeed is not technically 

 admissible to the purely kinematic and algebraic part of the subject proposed. But 

 it is not merely an ideal kinematic construction of the algebraic problem that is in- 

 tended ; and the design of a kinematic machine, for success in practice, essentially 

 involves dynamical considerations. In the present case some of the most important 

 of the purely algebraic questions concerned are very interestingly illustrated by 

 these dynamical considerations. 



