STATICS. 2.V) 



}'2'2'2. The necessary and sufficient conditions of equilibrium 

 of a system of forces acting on a rigid body are that the sum of 

 the moments of all the forces about each of the edges of any finite 

 tetrahedron shall be severally equal to zero. 



lL'123. Forces acting on a rigid body are represented by the 

 edges of a tetrahedron, three acting from one angular point and 

 the other three along the sides taken in order of the opposite 

 face ; prove that, whichever angular point be taken, the product 

 of the resultant force and minimum couple will be the same. 



li'24. Three forces act along three non-intersecting straight 

 lines, any other straight line is drawn meeting the three ; prove 

 that the shortest distance of this straight line from the central 

 axis of the forces is proportional to the cotangent of the angle its 

 direction makes with the central axis. 



]'2'25. A portion of a curve surface of continuous curvature 

 is cut off by a plane, and at a point in each element of the portion 

 a force proportional to the element is applied in direction of the 

 normal ; prove that, if all the forces act inwards, or all outwards, 

 they will in the limit have a single resultant. 



If a system of forces acting on a rigid body be reducible 

 to a couple, it is always possible by rotation about any proposed 

 point to bring the body into such a position that the forces, 

 acting at the same points of the body in the same directions in 

 space, shall produce equilibrium. 



1 'I'll. A system offerees are reduced t> a force acting through 

 MID.' 1 point and a couple; prove that, if the assumed point 

 be taken on a fixed straight line, and through it the axis of the 

 couple be drawn, the ,xi a will lie on an. 



fixed straight line. 



\--2S. Prove that the central axis of two forces P, Q in- 

 -tance c between their lines of action, 

 and divid I it in the ratio 



Q(Q+l'cn*0) : 7> (/> + <? cos 0), 



