314 Mr. stokes, ON THE FRICTION OF FLUIDS IN MOTION, 



in all directions. On this supposition we shall get the value of -y— from that of R' - K in 



dt 



du dv dw 1 dy^t 



the equations of page 140 by putting —- = -— = —- f= -^. We have therefore 



^ ' " • '^ ^ da: dy dss 3y_t dt 



dt 3 y^tdt 



Putting now for /3 + /3 its value 9.ak, and for — —~ its value given by equation (2), 



\t dt 



the expression for •ar, page 152, becomes 



idu dv dw\ 



a (du dv dw\ 



p + - {K + k) {— + ~ + — 

 ^ 3 ^ \dw dy d«j 



dy 



Observing that a{K + k) = (i, this value of -ay reduces Poisson's equations (9) to the 

 equations (12) of this paper. 



Poisson himself has not made this reduction of his equations, nor any equivalent one, so that 

 his equations, as he has left them, involve two arbitrary constants. The reduction of these two 

 to one depends on the assumption that a uniform expansion of any particle does not require a 

 re-arrangement of the molecules, as it leaves the pressure still equal in all directions. If we do 

 not make this assumption, but retain the two arbitrary constants, the equations will be the same 

 as those which would be obtained by the method of this paper, supposing the quantity k of 

 Art. 3 not to be zero. 



19. There is one hypothesis made in the common theory of elastic solids, the truth of 

 which appears to me very questionable. That hypothesis is the one to which I have already 

 alluded in Art. 17, respecting the legitimacy of neglecting the irregular part of the action of the 

 molecules in the immediate neighbourhood of the one considered, in comparison with the total 

 action of those more remote, which is regular. It is from this hypothesis that it follows as a 

 result that the molecules are not displaced among one another in an irregular manner, in 

 consequence of the directive action of neighbouring molecules. Now it is obvious that the 

 molecules of a fluid admit of being displaced among one another with great readiness. The 

 molecules of solids, or of most solids at any rate, must admit of new arrangements, for most solids 

 admit of being bent, permanently, without being broken. Are we then to suppose that when a 

 solid is constrained it has no tendency to relieve itself from the state of constraint, in consequence 

 of its molecules tending towards new relative positions, provided the amount of constraint be very 

 small .' It appears to me to be much more natural to suppose a priori that there should be some 

 such tendency. 



In the case of a uniform dilatation or contraction of a particle, a re-arrangement of its 

 molecules would be of little or no avail towards relieving it from constraint, and therefore it is 

 natural to suppose that in this case there is little or no tendency towards such a re-arrangement. 

 It is quite otherwise, however, in the case of what I have called a displacement of shifting. 

 Consequently B will be less than if there were no tendency to a re-arrangement. On the 

 hypothesis mentioned in this article, of which the absence of such tendency is a consequence, 

 I have said that a relation has been found between A and B, namely A = 5B. It is natural 

 then to expect to find the ratio of ^ to B greater than 5, approaching more nearly to 5 as the 

 solid considered is more hard and brittle, but differing materially from 5 for the softer solids, 

 especially such as Indian rubber, or, to take an extreme case, jelly. According to this view the 

 relation A = 5B belongs only to an ideal elastic solid, of which the solidity, or whatever we please 

 to call the property considered, is absolutely perfect. 



