AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 295. 



former changes with the time. In most cases to whicli it would be interesting to apply the 

 theory of the friction of fluids the density of the fluid is either constant, or may without sensible 

 error be regarded as constant, or else changes slowly with the time. In the first two cases the 

 results would be the same, and in the third case nearly the same, whether k were equal to zero or 

 not. Consequently, if theory and experiment should in such cases agree, the experiments must 

 not be regarded as confirming that part of the theory which relates to supposing k to he 

 equal to zero. 



4. It will be easy now to determine the oblique pressure, or resultant of the normal pressure 

 and tangential action, on any plane. Let us first consider a plane drawn through the point P 

 parallel to the plane yx. Let Ox^ make with the axes of .r, y, z angles whose cosines are I', m\ n' ; 

 let /", m", n" be the same for Oy,, and /'", m", n" the same for Ox^. Let P^ be the pressure, 

 and (xty'), {xtz) the resolved parts, parallel to y, x respectively, of the tangential force on the 

 plane considered, all referred to a unit of surface, {xty) being reckoned positive when the part 

 of the fluid towards — w urges that towards + a? in the positive direction of y, and similarlv 

 for (vtz). Consider the portion of the fluid comprised within a tetrahedron having its vertex 

 in the point P, its base parallel to the plane yz, and its three sides parallel to the planes w y , y x , 

 xx^ respectively. Let A be the area of the base, and therefore I' A, I" A, I'" A the areas of the faces 

 perpendicular to the axes of a?^, y., x^. By D'Alembert's principle, the pressures and tangential 

 actions on the faces of this tetrahedron, the moving forces arising from the external attractions, 

 not including the molecular forces, and forces equal and opposite to the effective moving forces will 

 be in equilibrium, and therefore the sums of the resolved parts of these forces in the directions 

 of .r, y and z will each be zero. Suppose now the dimensions of the tetrahedron indefinitely 

 diminished, then the resolved parts of the external, and of the effective moving forces will vary 

 ultimately as the cubes, and those of the pressures and tangential forces on the sides as the 

 squares of homologous lines. Dividing therefore the three equations arising from equating to zero 

 the resolved parts of the above forces by A, and taking the limit, we have 



Pi= 2/'- {p + p), {xty) = 2/' to' (p + p), (xtx) = ll'n' (p + p'), 



the sign 2 denoting the sum obtained by taking the quantities corresponding to the three axes 

 of extension in succession. Putting for js', p" and p"' their values given by (6), putting e'+e"+e"' 

 = sS, and observing that 2/'^= I, 'S.l'm'= 0, "S-tn = 0, the above equations become 



P,= p -2/ii2re' + 2m^5 (aity) = -2,x1l'm'e', (*<«)=- 2/i2/'wV. 



The method of determining the pressure on any plane from the pressures on three planes 

 at right angles to each other, which has just been given, has already been employed by MM. Cauchy 

 and Poisson. 



The most direct way of obtaining the values of 2i''^e' &c. would be to express I', m' and 

 n in terms of e by any two of equations (3), in which a;', y , z' are proportional to I', m, n', 

 together with the equation /'^ + m" + «'"= 1, and then to express the resulting symmetrical function 

 of the roots of the cubic equation (4) in terms of the coefficients. But this method would 

 be excessively laborious, and need not be resorted to. For after eliminating the angular motion of 

 the element of fluid considered the remaining velocities are e'x', e"y', e"'a/, parallel to the axes of 

 ■",) y,i ^,- The sum of the resolved parts of these parallel to the axis of x is /'e'.r_'+ l"e"y'+ l"'e"' z'. 

 Putting for .r ', y^', z' their values l'x'+ m'y + n z' &c., the above sum becomes 



.r'2i'°e' +y''^l'm'e' + z''2l'n' e ; 

 but this sum is the same thing as the velocity U in equation (2), and therefore we have 



)Ll''e'= -r- , 'S.I'm' e = ^ + ] , 2/'«V= J 7- + — • 



dx ^ \dy drj ^ \dx p.vl 



