80 REtORt— 1881. 



proportions of elasticity, plasticity, and viscosity. For viscous fluids, or 

 where B groups only are present, the equations of motion are found on 

 the supposition that in any element the groups not thrown out behave 

 as elastic solids, whilst those thrown out behave as perfect fluids, i.e. are 

 only in a state of contraction or dilatation, and that a strain of the element 

 is the sum of the strains of the first as an elastic solid, and of the second 

 as dilatations. The final equations obtained are : — 



with two others, where I is the ratio of the number of groups thrown out 

 per unit of time to those not thrown out, and r, h are the rigidity and 

 resistance to dilatation respectively for the elastic groups. With I large 

 or viscosity small, this becomes — 



T dd r ■. , /v- DwN f. 



whilst for I small (as in Canada Balsam), it is — 



Mr. Butcher also forms the equations of motion for plastic solids on 

 similar principles. 



Bobylew ' has transformed Stokes' form of the equations of motion of 

 a viscous fluid to curvilinear co-ordinates, and has given expressions for the 

 pressure at any point of a viscous fluid, and its rate of variation with the 

 time, analogous to those given by Helmholtz for a perfect fluid. Simpler 

 proofs of the same formulse have been given since by Forsyth ^ and Craig.' 

 When the motion at the boundary is zero, the rate of variation takes the 



simple form — 4^ vrdxctydz, where lo is the rotation at the point 



(x.y.z). It is clear, therefore, that with no motion at the boundary, or 

 in an infinite fluid at rest at infinity, there must be dissipation of the 

 energy of motion. Lipschitz * has also given expressions for the pressure 

 within a viscous fluid under the action of external and internal attraction. 

 None of these theories can be regarded as perfectly satisfactory ; even 

 Stokes', which has been most generally accepted, introduces stresses, 

 whose appearance, simply on the theory of friction acting on the surface 

 of an element of fluid, it is difiicult to understand. Take, for instance, 

 the motion given by i(,=ax, v=0, iv=0 in a compressible fluid ; this gives 

 a stress, due to friction, perpendicular to a small plane parallel to the 

 plane of yz, where certainly no force can arise from friction, if we 

 suppose it to act on the surfaces only. The method employed by Max- 

 well, and suggested by Stefan afterwards for extension to liquids, would 

 seeni the more hopeful road, but we must wait until the motions of 

 the molecules of liquids are better understood than at present. On a 

 > ' Einige Betrachtungen iiber die Gleichungen der Hydrodynamik,' Math. Ann. 



vi. p. 72. 



'' ' On the motion of a viscous incompressible fluid,' Mess. Math. ix. p. 134. 



' Journal of the Franhland Institute, October, 1880. Also < On certain possible 

 cases of steady motion in a viscous fluid,' Amer. Jour., iii. p. 269. 



* ' Determinazione della pressione nelF interne d'un fluido mcompressibile soggetto 

 ad attrazioni interne ed esterne,' BrioscM, Ann. (2) vi. 



