554 VISCOSITY. [CHAP, xi 



This equation, with 



* (vi), 



determines the values of a and m. Eliminating m, we find 



(a + i/F) {/o- 2 (a 2 + o- 2 ) - 4v 2 4 a 2 } 2 = M& 2 [a 2 {(a + 2^ 2 ) 2 + o- 2 } -/&amp;lt;r V] 2 . . .(vii), 



or, if we write 



a/o-=y, vk*/&amp;lt;r=0 ( y iii)&amp;gt; 



(y + 0) 1-f^jL (y2 +1) _ 40y\ = 0[~y 2 {(# + 20) 2 4a}-^/T....(ix). 



I CT J L &quot; J 



This equation has an extraneous root y = 0, and other roots are in 

 admissible as giving, when substituted in (v), negative values to the real part 

 of m. For all but very minute wave-lengths, 6 is a small number ; and, if we 

 neglect the squa re of 0, we obtain 



This is satisfied by y=i, approximately; and a closer approximation is 

 given by 



y(y 2 +l) 2 =0 (xi), 



leading to 



/ ^_\ . 0* 

 y ~ ~\ 2V2/ l 2^2 (XU) 



Hence, neglecting the small change in the speed of the oscillations, 

 The modulus of decay is therefore 



in the notation of Art. 246. 



Under the circumstances to which this formula applies the elasticity of the 

 oil-film has the effect of practically annulling the horizontal motion at the 

 surface. The dissipation is therefore (within limits) independent of the 

 precise value of /. 



The substitution of (x) for (ix) is permissible when 6 is small compared 

 with /o- 2 /cr 2 , or c small compared with fT /v. Assuming i/= 018, ^ =40, 

 we have T /v =2200. Hence the investigation applies to waves whose 

 velocity is small compared with 2200 centimetres per second. It appears on 

 examination that this condition is fulfilled for wave-lengths ranging from a 

 fraction of a millimetre to several metres. 



The ratio of the modulus (xiv) to the value (l/2i/& 2 ), obtained on the 

 hypothesis of constant surface-tension, is 4^/2 (^ 2 /o-)^, which is assumed to 

 be small. The above numerical data make X m =l 27, c m = 20. Substituting 

 in (xiv) we find 



r = -30X x(c ffi /c)*. 



For X=X m this gives r= 43sec. instead of 1 41 sec. as on the hypothesis 

 of constant tension. For larger values of X the change is greater. 



