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IntroductionGeneral remarks on the place of Quaternions in Physics Cartesian form of some of the results to follow Definitions Properties of ζ Fundamental Property of D Theorems in Integration Potentials Brief recapitulation of previous work in this branch Strain, Stress-force, Stress-couple Stress in terms of strain The equations of equilibrium Variation of temperatureSmall strains Isotropic Bodies Particular integral of the equation of equilibrium Orthogonal coordinatesSaint-Venant’s torsion problemWiresElectrostatics-general problemThe force in particular cases Nature of the stress Magnetism-magnetic potential, force, inductionMagnetic solenoids and shells Electro-magnetism-general theory Electro-magnetic stress PreliminaryNotation Euler’s equations The Lagrangian equationsCauchy’s integrals of these equations Flow, circulation, vortex-motion Irrotational Motion Motion of a solid through a liquid The velocity in terms of the convergences and spins ViscosityPreliminaryStatement of Sir Wm. Thomson’s and Prof. Hicks’s theories General considerations concerning these theories Description of the method here adopted Acceleration in terms of the convergences, their time-fluxes, and the spinsSir Wm. Thomson’s theory Prof. Hicks’s theoryConsideration of all the terms except −∇ (σ2)/2 Consideration of the term −∇ (σ2)/2