A Treatise on the Theory of Screws

CHAPTER VIII . THE POTENTIAL. 98. The Potential. Suppose a rigid body which possesses freedom of the ?ith order be sub- mitted to a system of forces. Let the symbol 0 define a position of the body from which the forces would be unable to disturb it. By a twist of amplitude O' about a screw 0 belonging to the screw system, the body may be displaced from 0 to an adjacent position P. the energy consumed in making the twist being denoted by the Potential V, and no kinetic energy being supposed to be acquired. The same energy would be required, whatever be the route by which the movement is made from 0 to P. So far as we are at present concerned V varies only with the changes of the position of P with respect to 0. The most natural co-ordinates by which the position P can be specified with respect to 0 are the co-ordinates of the twist (§ 32) by which the movement from 0 to P could be effected. In general these co-ordinates will be six in number; but if n of the screws of reference be selected from the screw system defining the freedom of the body, then (§ 95) there will be only n co-ordinates required, and these may be denoted by 0/, ... 0n'. The Potential V must therefore depend only upon certain quantities independent of the position and upon the n co-ordinates Ö/,... 6n'; and since these are small, it will be assumed that V must be capable of development in a series of ascending powers and products of the co-ordinates, whence we may write V = H + IIiei'+--+Hn0n' + terms of the second and higher orders, where II, H1, ... Hn are constants, in so far as different displacements are concerned. In the first place, it is manifest that H — 0; because if no displacement be made, no energy is consumed. In the second place, ... Hn must also be each zero, because the position 0 is one of equilibrium ; and therefore,