Cambridge Philosophical Society. 6 1 



axes of y and z, the residual relative motion depends on only six 

 independent quantities. Considering only this residual relative 

 motion, the author shows that there are always three directions, 

 which he names axes of extension, at right angles to one another, 

 such that if they be made the axes of x x , y,, z x , the resolved parts 

 of the relative velocity of the point P', whose relative co-ordinates 

 are #,, y,, z u will be e' x u e"y,, e'" z x along the three axes of exten- 

 sion respectively, the point P' being supposed indefinitely near to P. 

 Thus the system of pressures S is made to depend on the three 

 quantities e 1 , e", e'", which in the case of an incompressible fluid are 

 connected by the equation e' + e" + e'" = 0. Moreover, on account 

 of the symmetry of the motion, the pressures on planes perpendi- 

 cular to the axes of extension will be normal to those planes. They 

 will here be denoted by p', p", p'". 



By what precedes, any one of these pressures, as p', will be ex- 

 pressed by (e 1 , e", e'"), the function <p being symmetrical with re- 

 spect to the second and third variables. For reasons stated in the 

 paper itself, the author was led to take, as the form of the function 

 <p, £ <?' -f- £'(e" -f- e'"). The general expressions for the pressures 

 would thus contain two arbitrary constants (or rather functions of 

 the pressure and temperature), which in the case of an incompres- 

 sible fluid would unite into one. But it is shown by the author, 

 that in all probability /j'=0 when e'= e" = e'" ; and he accordingly 

 makes this assumption, which reduces the two constants to one, 

 even in the case of a gas. The expression for^' finally adopted is 



YP(e" = e'"-2e'). 



The pressures on three planes passing through P being known, the 

 pressure on any other plane passing through that point may be 

 found by the consideration of the motion of an indefinitely small 

 tetrahedron of the fluid. Thus expressions ■ are obtained for the 

 pressures on planes parallel to the co-ordinate planes. These ex- 

 pressions, however, contain quantities which refer to the axes of 

 extension ; and it is necessary to transform them into others con- 

 taining quantities which refer to the axes of co-ordinates. This 

 transformation is easily effected by means of an artifice, and then 

 no difficulty remains in forming the equations of motion. When p 

 is supposed to be constant, a supposition which it is shown may in 

 many cases be made, the equations thus obtained are those which 

 would be obtained from the common equations by subtracting 



(d 2 u cPu d q u\ [x, d /du dv dw\ 

 d7 i + df + J?) + Ydx \dx + ~dy + dz) 



dp 



from -r- in the first, and making similar changes in the other two. 



The particular conditions which must be satisfied at the boundaries 

 of the fluid are then considered, and the general equations applied 

 to a few simple cases. 



On considering these equations the author was led to observe, 

 that both Lagrange's and Poisson's proofs of the theorem that udx 



