18 Rev. J. Challis's Remarks on Lagrange's 



tained by a fixed point, or axis, or against a surface; and any, 

 consistent with its state of retention, when it is retained, just 

 as if it were acted upon by no forces at all. From this prin- 

 ciple it follows that the force or forces which produce the dis- 

 placement will at the first moment be solely employed in 

 moving an inert mass, and will not alter P, Q, R, &c. Hence 

 the tension of the cord will remain the same, and w will neither 

 ascend nor descend ; for any motion of w must be accompanied 

 by a change of tension. 



If then p be the interval between the blocks of the first 

 system of pulleys, q of the second, r of the third, &c. and / be 

 the length of the rest of the cord, its whole length == 



mp + m' q + m n r -f &c. + I. 

 This must remain the same whatever displacement be made. 

 Therefore 



mdp + m'lq + m"Zr -f &c. + $/ = 0, 

 or Vlp + Q8? + R8r + &c. + w8/ = 0, 

 whatever be the magnitudes of §p 9 8 q 9 8 r, &c. But if the dis- 

 placement be indefinitely small, it follows, from what is said 

 above, that 8 1 = 0. Consequently 



P8;? + Q8? + R8r + &c. = (A) 



If instead of a single mass, as we have supposed, the forces 

 P, Q, R, &c. acted on several masses connected by inexten- 

 sible cords or by hinges, for each of these masses an equation 

 like (A) will be obtained if the tensions of the cords and reac- 

 tions at the hinges be included in the forces. By the addition of 

 these several equations, the tensions and reactions will disap- 

 pear, because their virtual velocities enter with opposite signs. 

 The resulting equation will therefore still be of the form of (A). 

 If it be questioned how a method which seems to have no 

 reference to the first principles of statics, as given in the ele- 

 mentary treatises, should lead to a general solution of all sta- 

 tical problems, we may answer, that in the inductive method, 

 (as given for instance in M. Poisson's Treatise), only two prin- 

 ciples are admitted : 1°, that the direction of the resultant of 

 any two equal forces acting on a point bisects the angle which 

 the directions of the forces make with each other ; 2°, a force 

 produces the same effect at whatever point in its direction it 

 be applied. By the first it comes to pass that all equations 

 of equilibrium are homogeneous with respect to the forces ; 

 and such the general equation (A) becomes by the disappear- 

 ance of w ; by reason of the other, these equations are inde- 

 pendent of the distances of the points of application of the 

 forces from fixed points in their directions; and the same 



