OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOLUME XXI. Nos. III. AND IV. pp. 107-128. III. INTEGRAL FORMS AND THEIR CONNEXION WITH BY R. HARGREAVES, M.A. AND IV. ON THE APPLICATION OF INTEGRAL EQUATIONS TO THE BY H. BATEMAN, M.A., FELLOW OF TRINITY COLLEGE, CAMBRIDGE. CAMBRIDGE: AT THE UNIVERSITY PRESS. M.DCCCC.VIII. ADVERTISEMENT THE Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. THE SOCIETY takes this opportunity of expressing its grateful acknowledgments to the SYNDICS of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions. III. Integral Forms and their connexion with Physical Equations. By R. HARGREAVES. [Received in revised form May 7, 1908. Read May 18, 1908.] We are concerned here with the variation of integral forms, or more specially with their invariance, under the action of an operator which is an extension of the hydrodynamical operator in Euler's equations. In the integral forms temporal terms are admitted, i.e. terms containing dt as well as the differentials of coordinates, and it appears that the forms have special properties when the temporal terms are associated with the non-temporal in a definite way. These associated terms are significant quantities which include vector and scalar products as particular cases. The general theory comprises the action of the operator, its conjunction with the process of derivation in Stokes's theorem, and the law of association. The account of the general theory is followed by applications to the equations of hydrodynamics, to those of general dynamics, and to the electromagnetic equations. It is hoped that this may prove a useful contribution to the unification of the equations of physics on what may briefly be described as the principle of 'the invariance of circuital content.' In connexion with the significance of the circuit Stokes's theorem, in a generalized form, is of fundamental importance as revealing the quantities characteristic of an infinitesimal circuit, and also as furnishing the clue to the treatment of integrals for closed spaces. contains the same number of differentials, those in one term all distinct; the terms involve various combinations of the letters but not necessarily all the combinations of given order. The letters X and u denote functions of t and of the variables x. The operation DO/Dt is then understood to mean that ADVERTISEMENT THE Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. THE SOCIETY takes this opportunity of expressing its grateful acknowledgments to the SYNDICS of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions. III. Integral Forms and their connexion with Physical Equations. By R. HARGREAVES. [Received in revised form May 7, 1908. Read May 18, 1908.] We are concerned here with the variation of integral forms, or more specially with their invariance, under the action of an operator which is an extension of the hydrodynamical operator in Euler's equations. In the integral forms temporal terms are admitted, i.e. terms containing dt as well as the differentials of coordinates, and it appears that the forms have special properties when the temporal terms are associated with the non-temporal in a definite way. These associated terms are significant quantities which include vector and scalar products as particular cases. The general theory comprises the action of the operator, its conjunction with the process of derivation in Stokes's theorem, and the law of association. The account of the general theory is followed by applications to the equations of hydrodynamics, to those of general dynamics, and to the electromagnetic equations. It is hoped that this may prove a useful contribution to the unification of the equations of physics on what may briefly be described as the principle of the invariance of circuital content.' In connexion with the significance of the circuit Stokes's theorem, in a generalized form, is of fundamental importance as revealing the quantities characteristic of an infinitesimal circuit, and also as furnishing the clue to the treatment of integrals for closed spaces. contains the same number of differentials, those in one term all distinct; the terms involve various combinations of the letters but not necessarily all the combinations of given order. The letters X and u denote functions of t and of the variables x. The operation DO/Dt is then understood to mean that |