OE INTERNAL FEICTION OF AIE AND OTHER GASES. 259 



That part of the surface of the disk which is not near the edge may be treated as 

 part of an infinite disk, and we may assume that each horizontal stratum of the fluid 

 oscillates as a whole. In fact, if the motion of every part of each stratum can be 

 accounted for by the actions of the strata above and below it, there will be no mutual 

 action between the parts of the stratum, and therefore no relative motion between its 

 parts. 



Let 6 be the angle which defines the angular position of the stratum which is at the 

 distance y from the fixed disk, and let r be the distance of a point of that stratum from 



the axis, then its velocity will be r t^, and the tangential force on its lower surface arising 



from viscosity will be on unit of surface 



The tangential force on the upper surface will be 



/ dH . dH 



/ d^ , rf^fl , \ 

 ('''[dydt + dfdt^y)'^ 



and the mass of the stratum per unit of surface is §dy, so that the equation of motion of 



each stratum is 



dH d^ 

 ^W^=(^d^V (^) 



which is independent of r, showing that the stratum moves as a whole. 



The conditions to be satisfied are, that when y=0, ^=0 ; and that when y=b, 



dz=Ce-"cos{nt-\-o!.) (3) 



The disk is suspended by a wire whose elasticity of torsion is such that the moment 

 of torsion due to a torsion 6 is lu^d, where I is the moment of inertia of the disks. The 



di 

 r velocity 



The equation of motion of the disks is then 



viscosity of the wire is such that an angular velocity -j- is resisted by a moment 2Vc ~. 



l(S+< + -^^)+NAp^,=0, (4) 



where A=^2Tr^dr=^rr*, the moment of inertia of each surface, and N is the number of 

 surfaces exposed to friction of air. 



The equation for the motion of the air may be satisfied by the solution 



6=e-'*{e'"' cos (nt+qy)—e-'^ cos (nt—qy)}, (5) 



provided 

 and 



^M='p (6) 



I>'-f=fi (7) 



2n2 



