A Treatise on the Theory of Screws

103] THE POTENTIAL. 93 We now show that the expression of the potential will consist of the sum of n square terms, when referred to any set of n conjugate screws of the potential. 'I'he energy consumed in giving a body a twist of amplitude 0' from the position of equilibrium 0 to a new position P, is equal to Fv^ (5102) and v is the screw on which the wrench is evoked. Suppose that from the position P the body receive a twist of amplitude </>' about a screw <f>, it would generally not be correct to assert that the energy consumed in the second twist was proportional to the square of its amplitude. For during the second twist, either a portion of the energy will be consumed in doing work against the wrench on v, or the energy expended in the second twist will be rendered less, in consequence of the assistance afforded by the wrench on v. If, however, v be reciprocal to </>, then the quantity of energy con- sumed m the twist about </> will be unaffected by the presence of a wrench on y. Hence if 0 and </> be two conjugate screws of the potential, the energy expended in giving the body first a twist of amplitude 0' about 0, and then a twist of amplitude </>' about (f>, is to be represented by Fv^0'-+ Fv^dP. By taking a third screw, conjugate to both 0 and </>, and then a fourth screw conjugate to the remaining three, and so on, we see finally that the potential reduces to the sum of n square terms, where each pair of the screws of reference are conjugate screws of the potential.