IN GEOMETRY, MECHANICS, AND ASTRONOMY. 421 



(II). 



19. To deduce the equations (28) and (29) immediately from the parallelogram of forces. 

 We must premise the following Lemmas. 



20. Lemma 1. If u and u be any two points situated on the line of direction of the force 

 U, then Du .U= Du'. U. 



For the line {u - u)* coincides in direction witli the line U (forces being supposed to be 

 represented by lines) ; and it is therefore evident from Art. 5, that D{u'- ti) . U = 0, i.e. Du'. U 

 = Die. U. 



21. Lemma 2. If three forces P, Q, B, applied to a rigid body at the points p, q, r respec- 

 tively, balance each other, then the conditions of equilibrium are 



P + Q + .ff = (30). 



Dp.P + Dq.Q+ Dr. R = (si). 



For /', Q, and R must meet in the same point ; let ?< be that point : also - R must be the 

 resultant of P and Q, and therefore, expressing the parallelogram of forces symbolically, we have, 



- R = P + Q, or, 



P+ Q + R = 0. 



Now performing the operation Du . on this equation, we have 



Du .P+ Du. Q + Du.R = 0, 

 and therefore, by Lemma 1, 



Dp.P+ D(j .Q + Dr.R =: 0. 



Hence the conditions (30) and (31) must hold if P, Q, and R balance each other. 



And, conversely, if (30) and (31) be true, the forces P, Q, and R will balance each other. For 

 let u be the point of intersection of P and Q; then, by Lemma 1, we have Du . P = Dp . P, and 

 Du . Q = Dq . Q; and therefore by (31), we have 



Dr.R =^ - Du.{P + Q) = Du.R, by (30). 



Hence D (r - u) . R = 0, and therefore the line r - u coincides with R in direction, i.e. u is a 

 point in the line of direction of R. Hence P, Q, and R meet in the same point n. Also by (30), 



- R = P + Q, i.e. - i? is the resultant of P and Q. Hence P, Q, and R balance each other if 

 the conditions (30) and (31) be satisfied. These conditions therefore are necessary and sufficient for 

 (.quililjrium. 



22. From these Lemmas we may now prove that the equations (28) and (2y) are the neces- 

 sary and sufficient conditions of equilibrium of a rigid body, acted upon by the forces U, U', U", 

 8ic. at the points u, h, u", &c. 



Choose any three points^, p, q, r, in the rigid body ; resolve U into three forces acting along 

 the lines u - p, u - q, u - r, (i.e. the lines drawn from p, q, and r to m) ; let P, Q, R denote 

 these forces respectively ; in like manner resolve U' into P\ Q', R\ acting respectively along tlie 

 lines u - p, u - q, u' — r: treat IJ" similarly, and so on. 



Then the forces U, U , U", &c. are reduced to the three sets of forces, 



P, P, P", &.C. acting at the point p, 



Q, Q', Q", &C q, 



R, R', R", &.C r. 



• (u'-u) exprcnnes in inaBniliide and dircclioii the line drawn from the point » to the point u'. 

 + ThcHC points are Kuppostd not to lie in the sume right line. 



