OB INTERNAL FEICTION OF AIR AND OTHER GASES. 261 



of an infinite plane parallel to xz, oscillating in the direction of z between two planes 

 infinite in every direction at distance b. Let w be the velocity of the fluid in the direc- 

 tion of z, then the equation of motion is* 



dw (cPw d^w\ 



sw=i^[di''+w)' ^ ^^ 



with the conditions 



w=0 ■wh.eny=+b, (11) 



and 



w=Ccosnt 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 between the disks is 



,?„ 



so small, that we may neglect — ^-, and the equation is reduced to 



£+$=0 (13) 



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

 to Professor W. Thomsok, who has shown me how to transform these conditions into 

 another set with which we are more familar, namely, w=0 when a;=0, and w=l 

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



w=^sin-'!ip, (14) 



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

 If we put 



P=IH {\/(^i+^.)'-4+r,+rJ, (15) 



then the lines for which p is constant will be ellipses orthogonal to the hyperbolas, and 



2+^t=0; (16) 



and the resultant of the friction on any arc of a curve will be proportional to ip,— fo, 



where ^o is the value of p at the beginning, and ^, at the end of the given arc. 



2 2 



In the plane y=0, when x is very great, <p=- log 4x, and when x=l, <p= ~ log 2, 



2 

 so that the whole friction between ^•=1 and a very distant point is - log 2x. 



Now let w and (p be expressed in terms of r and S, the polar coordinates with respect 

 to the origin as the pole ; then the conditions may be stated thus : 



* Professor Stokes " On the Theories of the Internal Friction of Fluids in Motion, &c.," Cambridge Phil. 

 Trans, vol. viii. 



