In the special case of two forces, we see that must coincide with 0, if the forces are to be in equilibrium. Clearly, this can only occur when the forces are equal and opposite, with the same line of action.
On the other hand, the distor tion is generally small, and for many purposes it may be neglected. Statics commonly treats of bodies as rigid, and it makes use of an assumption which can be regarded as intuitive, viz., that any point in a body, lying on the line of action of a force, may be regarded indifferently as the point of application. This is the principle of transmissibility of force; it enables us, in effect, to concentrate upon forces, without particular reference to the body upon which they act.
42. Thus, if three forces combine to maintain equilibrium in a body of any size, we may assert, quite generally, that they must be concurrent. This theorem will be seen to follow directly from the condition for two forces which was stated at the end of § 38: the third force must be equal and opposite to the resultant of the other two, and therefore it must act through their point of inter section. As fig. 14 indicates, the point at which three forces inter sect will not necessarily fall within the body upon which they act.
Two cases demand examination, shown in diagrams (A) and (B), respectively, of fig. 15. In the first case, P and Q have the same sense, and C lies between A and B: in the second, P and Q are opposite in sense, and C lies outside AB, on the side of the greater force P. In either case we have The magnitude of the resultant force is evidently (P+Q) in the first case and (P—Q) in the second; its sense is in both cases that of i.e., the greater of the forces P and Q.
Thus one problem is solved, except in the special case in which P and Q are equal in magnitude and opposite in sense. In this case, according to (54), the point C is an infinite distance from A and B, and the magnitude of the resultant is zero : in effect, we cannot find a single force which will replace a pair of equal and opposite parallel forces whose lines of action are not coincident. A pair of forces of this nature is said to constitute a couple.