260 ME. CLEEK MAXWELL ON THE VISCOSITY 



and in order to fulfil the conditions (3) and (4), 



2ln(l-k)(e'"''-\-e-""'-2 cos 2qb)='^A[j>,{(pn-Iq)(e^-e-"") + (gn+lp)2 sin 2qb}. . (8) 



Expanding the exponential and circular functions, we find 



2lb(l-k}=^AiJi,{l-,ld+l<^in'-BP)+,\Mn'l-n+iioC%l6n*+^^^^ 



where c= — ^» 



I = observed Napierian logarithmic decrement of the amplitude in unit of time, 

 k=: the part of the decrement due to the viscosity of the wire. 



When the oscillations are slow as in these experiments, when the disks are near one 

 another, and when the density is small and the viscosity large, the series on the right- 

 hand side of the equation is rapidly convergent. 



When the time from rest to rest was thirty-six seconds, and the interval between the 

 disks 1 inch, then for air of pressure 29*9 inches, the successive terms of the series were 



1-0-0-00508 +0-24866 +0-00072 +0-00386=1-24816; 



but when the pressure was reduced to 1-44 inch, the series became 



l-0_0-0002448 +-0005768 + -00000008 +00000002=1-0003321. 



The series is also made convergent by diminishing the distance between the disks. 

 When the distance was -1847 inch, the first two terms only were sensible. When the 

 pressure was 29-29, the series was 



l_-000858 +-000278=1--00058. 



At smaller pressures the series became sensibly =1. 



The motion of the air between the two disks is represented in fig. 8, where the upper 

 disk is supposed fixed and the lower one oscillates. A row of particles of air which 

 when at rest form a straight line perpendicular to the disks, will when in motion assume 

 in succession the forms of the curves 1, 2, 3, 4, 5, 6. If the ratio of the density to the 

 viscosity of the air is very small, or if the time of oscillation is very great, or if the 

 interval between the disks is very small, these curves approach more and more nearly to 

 the form of straight lines. 



The chief mathematical difficulty in treating the case of the moving disks arises from the 

 necessity of determining the motion of the air in the neighbourhood of the edge of the disk. 

 If the disk were accompanied in its motion by an indefinite plane ring surrounding it 

 and forming a continuation of its surface, the motion of the air would be the same as if 

 the disk were of indefinite extent ; but if the ring were removed, the motion of the air 

 in the neighbourhood of the edge would be diminished, and therefore the efiect of its 

 viscosity on the parts of the disk near the edge would be increased. The actual effect 

 of the air on the disk may be considered equal to that on a disk of greater radius 

 forming part of an infinite plane. 



Since the correction we have to consider is confined to the space immediately 

 surrounding the edge of the disk, we may treat the edge as if it were the straight edge 



