■ilO Mr. stokes, on THE FRICTION OF FLUIDS IN MOTION, 



quantities. Substituting in the above equations, and observing that ^my'z' — ^m'z\v' 

 = Sm.r'y' = 0, and 2»i.r'- = 'S.my' = Swsr'^, we have 



/dw dv 



(dw av\ 



\dy 



We see then that an indefinitely small element of the fluid, of which the three principal moments 

 about the centre of gravity are equal, if suddenly solidified and detached from the rest of 

 the fluid will begin to move with a motion simply of translation, which may however vanish, 

 or a motion of translation combined with one of rotation, according as udx + vdy + todz is, 

 or is not an exact differential, and in the latter case the angular velocities will be the same 

 as in Art. 2. 



The principle which forms the subject of this section might be proved, at least in the case 

 of a homogeneous incompressible fluid, by considering the change in the motion of a spherical 

 element of the fluid in the indefinitely small time dt. This method of proving the principle 

 would show distinctly its intimate connexion with the hypothesis of normal pressure, or the 

 equivalent hypothesis of the equality of pressure in all directions, since the proof depends on 

 the impossibility of an angular velocity being generated in the element in the indefinitely small 

 time df by the pressure of the surrounding fluid, inasmuch as the direction of the pressure at 

 any point of the surface ultimately passes through the centre of the sphere. The proof I 

 speak of is however less simple than tiie one already given, and would lead me too far from 

 my subject. 



SECTION III. 



Application of a method analogous to that of Sect. I. to the detennmation of the equations 

 of equilibrium and motion of elastic solids. 



15. All solid bodies are more or less elastic, as is shown by the capability they possess 

 of transmitting sound, and vibratory motions in general. The solids considered in this section 

 are supposed to be homogeneous and uncrystallized, so that when in their natural state the 

 average arrangement of their particles is the same at one point as at another, and the same 

 in one direction as in another. The natural state will be taken to be that in which no forces 

 act on them, from which it may be shown that the pressure in the interior is zero at all 

 points and in all directions, neglecting the small pressure depending on attractions of the 

 nature of capillary attraction. 



Let ,T, y, X be the co-ordinates of any point P in the solid considered when in its natural 

 state, a, /3, 7 the increments of those co-ordinates at the time considered, whether the body 

 be in a state of constrained equilibrium or of motion. It will be supposed that a, /3 and y 

 are so small that their squares and products may be neglected. All the theorems proved in 

 Art. 2. with reference to linear and angular velocities will be true here with reference to linear 

 and angular displacements, since these two sets of quantities are resolved according to the same 

 laws, as long as the angular displacements are supposed to be very small. Thus, the most 

 general displacement of a very small element of the solid consists of a displacement of translation, 

 an angular displacement, and three displacements of extension in the direction of three rectangular 

 axes, which may be called in this case, with more propriety than in the former, axes of 

 extensioti. The three displacements of extension may be resolved into two displacements of 

 shifting, each in two dimensions, and a displacement of uniform dilatation, positive or negative. 

 The pressures about the element considered will depend on the displacements of extension only; 



