A Historical Note on the Reciprocal Theorem and Theory of Statically Indeterminate Frameworks

Abstract

OPINIONS seem to vary concerning the origin of the reciprocal theorem. Thus, in his book “Mathematical Theory of Elasticity” (4th edit. 1927), A. E. H. Love attributed the theorem to E. Betti (Il nuovo Cimento, ser. 2, 1872), but stated that it is a special case of a more general theorem due to Lord Rayleigh (London Math. Soc. Proc., 4, 1873). In his book “Theory of Sound” (Vol. 1, 2nd edit. 1894), however, Rayleigh also acknowledged the reciprocal theorem as due to Betti. Neither Love nor Rayleigh connected Maxwell's name with the theorem although it is widely known as “Maxwell's Reciprocal Theorem”. In fact, Maxwell did include a statement of a form of the reciprocal theorem in his paper “On the Calculation of the Equilibrium and Stiffness of Frames” (Phil. Mag., 27, 1864). It is, no doubt, due to this work, which appeared before the relevant work of Betti and Rayleigh, that Maxwell is nowadays given credit for the theorem, and it is interesting to note the neglect of Maxwell's paper by Rayleigh and Love.