Proof of the Principle of Virtual Velocities. 19 



tiling happens in the equation (A) on account of the indefi- 

 nitely small magnitudes of $p, § q,$r, &c. 



I will here add an example of applying the equation (A) in 

 a manner which may be in some respects new. Three forces 

 P, Q, R, act in the same plane on a rigid rod; xy f x'i/, x" i/ 1 

 are the coordinates of their points of application referred to 

 rectangular axes in the plane ; and 0, 0', 0" are the angles 

 which the directions of P, Q, R, make with the axis of x. 

 Now whatever small displacement be given to the rod, it may 

 be considered to be produced by its revolving through a small 

 angle $ A about some fixed point in the plane. Let the co- 

 ordinates of this point be X, Y ; and let its distances from 

 the points of application of the forces be r, r\ r". Then 

 these points move through r 8 A, r 1 8 A, r" 8 A. If a, a', a" 

 be the angles which r, r', t Jf make with the axis of x 9 then 



\ + * — *, ^- + ^ — 0', -|- + a" — 0" are the angles which 



the directions of the forces make with the directions of the 

 motions of their points of application. Hence the virtual ve- 

 locities are r 8 A cos (~ + u — \ r 1 $ A cos ( ~ + a! — 0'), 



and r" 8 A cos ( ~ + a" — 0" j . Therefore by the equation (A), 



Prsin(a-0) + Q H sin (et' - 0') + Rr"sin (a" - 0") = 0. 



But sin * = * , cos a = , &c. 



r r 



Hence = P {(y - Y) cos0 - (x - X) sin 0} 



+ Q{(i/- Y)cos0- (y- X)sin0} 



+ R {(/ - Y) cos - (x" - X) sin 0} 



As this equation is to be true whatever be the displacement, 

 that is, whatever be X and Y, we must have, 



P cos + Q cos 0' -|- R cos 0" = 

 P sin + Q sin 0' + R sin 0" = 

 P (y cos — x sin 0) + Q (y' cos 0' — x 1 sin 0') 

 + R(y , cos0" - .r" sin 0") = 0, 

 the known equations applicable to this instance. 



If the point of application of R be fixed, the displacement 

 must consist in a motion of rotation about this point. Making 

 it the origin of coordinates, we havey = 0, x" = 0, Y = 0, 

 X = 0, and P (y cos — x sin 0) + Q (3/ cos — x 1 sin 0) = 0, 

 which is the equation of equilibrium on the lever. 

 Pap worth St. Evcrard, Nov. 16, 1832. 



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