16 



ON THE VISCOSITY OR INTERNAL FRICTION 



the straight edge of an infinite plane parallel to xz, oscillating in the direction 

 of s between two planes infinite in every direction at distance b. Let w be 

 the velocity of the fluid in the direction of z, then the equation of motion is* 



dto d*w <fw\ 



with the conditions 

 and 



. . 



(10 >' 



w=Q when y=b .............................. (11), 



w = Ccos nt when y = 0, and x is positive. ...(12). 



I have not succeeded in finding the solution of the equation as it stands, 

 but in the actual experiments the time of oscillation is so long, and the space 



1 1 



between the disks is so small, that we may neglect - - , and the equation 

 is reduced to 



(13) 



with the same conditions. For the method of treating these conditions I am 

 indebted to Professor W. Thomson,, who has shewn me how to transform 

 these conditions into another set with which we are more familiar, namely, 

 u> = when x = 0, and w=l when y = 0, and x is greater than +1, and w= 1 

 when x is less than 1. In this case we know that the lines of equal values 

 of w are hyperbolas, having their foci at the points y = 0, x=l, and that 

 the solution of the equation is 



u>= -sin" 

 rr 



where r,, r, are the distances from the foci. 



If we put 



then the lines for which 

 hyperbolas, and 



= - log y(r, + r,)' - 4 + r, + r,}. 



TT 



(14) 



.(15), 



is constant will be ellipses orthogonal to the 



.(16); 



and the resultant of the friction on -any -arc of a curve will be proportional 



* Professor Stokes " On the Theories of the Internal Friction of Fluids in Motion, <fec.," Cambridge 

 Phil. Traru. VoL Tin. 



