HOGG. — FRICTION AND FORCE DUE TO TRANSPIRATION. 121 



made regarding the resistance which the gas in contact with the solid, 

 C D, offers to the motion of C D. Under these circumstances, in any 

 second, any one layer must move with respect to the next layer helow 

 it exactly the same distance which the one above it moves with respect 

 to the layer in question. It follows, then, that the line of particles, 

 X Y, at the beginning of a second will become the line X at the end 

 of that second. 



If, however, the solid and the gas in contact with it have different 

 velocities, that is, if the gas slips on the solid, the line X' Y' wiU rep- 

 resent the position of the line X Y at the end of the second. It is 

 clear from the figure that if we remove the planes farther from each 

 other by the distance P plus Y P', that is, twice P, and assume 

 that there is no slipping at the surface of the solids, then, since the 

 relative positions of the successive layers are unaltered, the friction be- 

 tween contiguous layers of gas, and therefore the tangential stress on 

 the solids will be the same as it is where there is slipping and where the 

 planes are at the original distance X Y from each other. 



That the distance P is, as Maxwell showed, equal to /u,/o-, is readily 

 seen. For, if we assume that the surfaces of the fixed and moving 

 solids are alike, o- will be the same for both. If, also, we call the dis- 

 tance Y Y', or X' 0, X, the tangential stress at either solid must be 

 (Tx, since x is the velocity of CD relative to the gas in contact with 

 it, and also the velocity of the gas in contact with A B, relative to 

 A B. But the stress between consecutive layers is, since the distance 

 between the planes is one centimeter, equal to yu, (1 — 2 .r), and we have 

 seen that this is the same as the stress at the solid. We have, then, 



<TX = ^ (\ — 2x) 



or, X = 



(TX 



2/A -f o- 



(T fX. 



2/-(. -j- o- 



= the force at the solid. 



But the form of this result shows that this is the force which must be 

 applied to the solid, C D, to maintain a velocity of one centimeter per 

 second if the distance from the fixed solid is (2 /u/cr-f 1) and there is 



