262 ME. CLEEK MAXWELL ON' THE VISCOSITY 



When 0=+ 2 J ^=0. When ^=0 and r greater than 1, w=l. When 6=x and r 



greater than 1, w= — 1. 



Now let x', y be rectangular coordinates, and let 



2 2 



q/z^-M and 3[/=.-b\oQr, (17) 



and let w and (p be expressed in terms of a/ and y ; the differential equations (13) and 

 (16) wiU still be true; and when y=i^j w=0, and when y = and a/ positive, w=l, 



3^ 2 2 



When x' is great, <P=-i +- log 4, and when a;=0, (p=- log 2, so that the whole friction 



on the surface is 



f^+^«g2, (18) 



26 

 which is the same as if a portion whose breadth is — log, 2 had been added to the surface 



at its edge. 



The curves of equal velocity are represented in fig. 9 at u, v, w, x, y. They pass 

 round the edge of the moving disk A B, and have a set of asymptotes U, V, W, X, Y, 

 arranged at equal distances parallel to the disks. 



The curves of equal friction are represented at o, ^, j, r, s, t. The form of these 

 curves approximates to that of straight lines as we pass to the left of the edge of the 

 disk. 



The dotted vertical straight lines O, P, Q, E, S, T represent the position of the cor- 

 responding lines of equal friction if the disk A B had been accompanied by an extension 

 of its surface in the direction of B. The total friction on A B, or on any of the curves 

 u, V, w. Sec, is equal to that on a surface extending to the point C, on the supposition 

 that the moving surface has an accompanying surface which completes the infinite plane. 



In the actual case the moving disk is not a mere surface, but a plate of a certain 

 thickness terminated by a slightly rounded edge. Its section may therefore be compared 

 to the curve uu' rather than to the axis A B. 



The total friction on the curve is still equal to that on a straight line extending to 

 C, but the velocity corresponding to the curve u is less than that corresponding to the 

 line A B. 



If the thickness of the disk is 2j3, and the distance between the fixed disks =2i, so 

 that the distance of the surfaces is b—(3, the breadth of the strip which must be sup- 

 posed to be added to the surface at the edge will be 



a=^log.l0|log.„2 + log,.sm'^j*. (19) 



In calculating the moment of friction on this strip, we must suppose it to be at the 



* This result is applicable to the calculation of the electrical capacity of a condenser in the form of a disk 

 between two larger disks at equal distance from it. 



