508 VISCOSITY. [CHAP. XI 



As in Art. 198 we could easily write down the expressions for the forced 

 oscillations in the general case where Q lt $ 2 vary as e i&amp;lt;T \ but we shall here 

 consider more particularly the case where &amp;lt;? 2 = and $ 2 = 0. The equations (i) 

 then give 



ia (6 19 8) &amp;lt;?i + (i&amp;lt;ra 9 + b 99 ) 6 9 = 



TT - Vl^r /ii x Z v / 29 



Hence 



n ,. 



A 2) &quot; 2 2 2 



This may also be written 



a,a 2 {(iV + a^ + oY 2 } (tcr + a,) 



11 



Our main object is to examine the case of a disturbing force of long period, 

 for the sake of its bearing on Laplace s argument as to the fortnightly tide 

 (Art. 210). We will therefore suppose that the ratio o-j/tr, as well as o^/c^, is 

 large. The formula then reduces to 



i&amp;lt;ra. 2 +b 22 



Everything now turns on the values of the ratios o-/a 2 and aa.Jb^. If o- be so 

 small that these may be both neglected, we have 



in agreement with the equilibrium theory. The assumption here made 

 is that the period of the imposed force is long compared with the time in 

 which free motions would, owing to friction, fall to e~ of their initial 

 amplitudes. This condition is evidently far from being fulfilled in the case of 

 the fortnightly tide. If, as is more in agreement with the actual state of 

 things, we assume o-/a 2 and o-a 2 /6 22 to be large, we obtain 



as in Art. 198 (vii). 



Viscosity. 



281. sAVe proceed to consider the special kind of resistance 

 which is met with in fluids. The methods we shall employ are of 

 necessity the same as are applicable to the resistance to distortion, 

 known as elasticity, which is characteristic of solid bodies. 

 The two classes of phenomena are of course physically distinct, 

 the latter depending on the actual changes of shape produced, 

 the former on the rate of change of shape, but the mathema 

 tical methods appropriate to them are to a great extent identical. 



If we imagine three planes to be drawn through any point P 



