152 STATICS. [247. 



Let m, m 1 be the lengths of the diagonals, T, T' the tensions, and 

 $m, 8m' the changes of length of the diagonals when the parallelogram 

 is slightly deformed ; then by the principle of virtual work 



o. (i) 



From geometry we have, if a, b are the sides of the parallelogram, 



hence, differentiating, m 8m + m' 8m' = o. (2) 



From (i) and (2) we find 



7/7" = */*'. (3) 



247. For the purposes of statics a body is regarded as rigid 

 if the points of application of all the forces acting on the body 

 have invariable distances from each other; these points may 

 be imagined connected by a framework of rigid rods. The 

 tensions in these connecting rods, since they occur in pairs of 

 equal and opposite forces acting along rigid lines, do not enter 

 into the equation of virtual work. One of the chief advantages 

 of the principle of virtual work consists in this elimination of 

 these internal forces. 



Let us now generalize the idea of the rigid body by assuming^ 

 the points of application of the forces to be connected by rods- 

 or threads which may even be elastic ; the points may also be 

 constrained to move on smooth surfaces or curves ; friction, 

 however, is to be excluded. 



Let x, y, z be the co-ordinates of one of the points, P \ 

 x' t _/, z' t those of another point, Q ; let / be the length of the 

 connecting thread or rod, PQ ; , ft, 7, its direction cosines ; 

 and let T be the tension or stress in PQ. If the whole system 

 be subjected to any infinitesimal displacement, for which &r, fry, 

 Sz are the component displacements of P, 8x', S/, Bz r those of 

 Q, the sum of the works of the two equal and opposite forces T 

 for this displacement will be 



To. 



