OF AIR AND OTHER GASES. 17 



to ^-^o, where <j> is the value of <f> at the beginning, and <, at the end 

 of the given arc. 



In the plane y = 0, when x is very great, <i = -10240;, and when x=l 



TT 



2 



-log 2, so that the whole friction between x=l and a very distant point 



is -Iog2x. 

 ti 



Now let w and < be expressed in terms of r and 6, the polar co-ordinates 

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



When 9=^, w = 0. When = and r greater than 1, w?=l. When 



Zi 



6 = TT and r greater than 1, w= 1. 



Now let x', y' be rectangular co-ordinates, and let 



2 2 



y' = -b& and a;' = -61ogr ........................ (17), 



IT TT 



and let w and <f> be expressed in terms of x and i/ ; the differential equa- 

 tions (13) and (16) will still be true; and when y'=b, w = 0, and when 

 y* = Q and x' positive, w=l. 



x' 2 2 



When x' is great, < = 7- + -log4, and when x' = 0, < = -log2, so that the 



TT TT 



whole friction on the surface is 



which is the same as if a portion whose breadth is : - log, 2 had been added 



TT 



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 AB, and have a set of asymp- 

 totes U, V, W, X, Y, arranged at equal distances parallel to the disks. 



The curves of equal friction are represented at o, p, q, 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. 



VOL. n. 3 



