728 MECHANICS These equations give 6, <p, , . . as homogeneous linear functions of /dT fdT .... , /dA fdA T , I j? ) i ( JT ) i that is, of I -r- , I -r- ) , ... ihus, d& J d<f>J dd J d<f>J if we substitute these expressions in the equation which is obviously true, because T is a homogeneous function of 6, 0, ... of the second degree, we have a partial differential equation of the form /rfAV r U; + = 2(H - V) f;om which A is to be found. The coefficients p, q, r . . . are, in general, like P, Q, R, . . . functions of 0, <p, . . . As an illustration, take again the example in last section, where two masses are attached to a helical spring, and vibrate in a vertical line. From the value of T there given we have s From these we have the equation for A The value of V is given above. This equation is, of course, to be treated according to the process illustrated in 209. STATICS OP A RIGID SOLID. Equili- 2 1 7. A rigid body, as we have already seen, has at the brium of utmost six degrees of freedom, three of translation and three of rotation. According to Newton s scholium, the conditions of equilibrium of such a body, under the action of any system of forces, are that the algebraic sum of the rates of doing work by and against the forces shall be nil whatever uniform velocity of translation or of rotation the body may have. For, if this were not so, there would be work done against acceleration, and the body would gain or lose kinetic energy. And this gain or loss would take place even if the body were originally at rest, i.e., it would not be in equilibrium. To ensure equilibrium then, all that is necessary is that the sums of the components of the forces in any three non-coplanar directions shall vanish, along with the sums of their moments about any three non-coplanar lines. For simplicity it is usual to assume for these direc tions a system of rectangular axes, and for the lines another system parallel to them and passing through some definite point (say the centre of inertia) of the body. The six Thus we have at once 2(X) = 0, 2(Y) = 0, 2(Z) = 0; where X, Y, Z are the components, parallel to the axes, of a force acting at the point x, y, z of the body. If P, with direction cosines A, p, v, represent the force acting at x, y, z, these equations may be written in the form 2(PA) = 0, 2(P/0 = 0, 2(PiO = These equations correspond to the six degrees of freedom involved. It is easy to see that it is a more matter of convenience through what point of the body we draw the lines about which moments are taken. For, if we shift it by quantities a, b, c respectively, the moments become but these are 2(Zy - Ys) - &2(Z) + c2(Y) , &c. ; and, by the first three equations, these quantities are seen to reduce themselves to their first terms. Hence, in forming the equations of equilibrium, simplicity will be gained by choosing as origin a point through which the line of action of one or more of the applied forces passes. Again, the point of application of any one of the forces may be shifted at will anywhere along the line in which the force acts. For the equations of the line in which the force at x, y, z acts are x - x if - y z - z X " Y " Z and these give Zy - Ys = J so that the expressions for the moments are unaltered if the point of application of the force be shifted to any position along the line in which it acts. 218. In the great majority of treatises on Statics the fundamental propositions of the subject, above given, are deduced from the assumption (as a thing to be proved ex perimentally) of the result just established, which is desig nated the "principle of the transmission of force." Along with it are assumed the parallelogram of forces, and the principle of the "superposition of systems of forces in equilibrium." Since the publication of the Principia, the continued use of such methods must be looked upon as a retrograde step in science. 219. From this category we cannot quite except (so far Couple, at least as the usual modes of treating it are concerned) the valuable idea of the " couple," due to Poinsot. But the term is in such common use, and the idea in its applica tions sometimes of such importance, that it cannot be omitted here. A couple is a pair of equal forces acting on the same body in opposite directions and in parallel lines. From the general conditions already given we see that a couple produces a definite moment of force about a particu lar axis, but that the axis is determinate merely as regards direction, and not as regards position in space. The forces of a couple do not appear in the first three of the equa tions of equilibrium. On the other hand, the left hand members of the other three equations may all be regarded as moments of couples. All the properties of couples are contained in these statements. Thus, for instance, it is obvious that, so far as its effects are concerned 1. A couple may be shifted by translation to any other Trans position in its own plane. ferenceo 2. It may bs shifted to any parallel plane. couple. 3. In either of these it may be turned through any angle. 4. Its forces may be increased or diminished in any Ami of ratio, provided the distance between their lines of action couple, (which is called the " arm " of the couple) be proportion ately diminished or increased. A couple is therefore completely determined by means Axis of of its " axis," which is a line drawn perpendicular to its couple, plane, and of length representing its moment. And two couples are obviously to be compounded by treating their axes as if they were forces acting at one point. 220. We will now examine the consequences of the Reduc- six conditions of equilibrium ( 217) in some of the more tion ot common cases which present themselves. But, before a s ^ ttl doing so, it may make matters clearer if we restate these an( j ^ 1L conditions in a somewhat different form. couple. The resultant of any number of forces, acting at any points of a rigid body, may be represented by a single force acting at the origin, and a couple of definite moment about a definite line passing through the origin. For equilibrium of the body this force and couple m*;st separately vanish. Thus if, in fig. 59, P, acting at Q, be any one of the forces, and O the origin (chosen at random), we may introduce at O a pair of equal and opposite forces P, parallel to P. The original force, taken along- with P at the origin, gives a couple ; and in addition there is + P acting at the origin.
221. When only two forces act on a body, the first