206] SLOW MOTIONS. 555 



If this is to be periodic in t, n must be pure imaginary, say 

 n = ip. 



Then we have 



and 



of which complex quantity both the real and the imaginary parts must 

 separately satisfy the equation 258), when multiplied by arbitrary 

 constants. Let us accordingly take 



259) u = | A cos (pt y -j- y) + J5sin (pt 



\ \ i A r / \ 



This represents a wave of frequency ~^ and wave-length 2jrl/- 



travelling with velocity y%vp, which as we see varies as the square 

 root of the frequency. Unlike our waves in perfect fluids however 

 it falls off in amplitude, being rapidly damped as we go into the 



fluid, being reduced in the ratio e~ ** = in each wave-length. 



Thus such motions are propagated but a short distance into a fluid. 

 In a similar manner the absorption of light by non- transparent media 

 is explained, the ether there having the properties of a viscous solid. 

 If we treat the equations 233) in the same way as we did 27) 

 in obtaining equation 57), 191, we obtain instead the following, 



d /|\ &du ndu du <. 



di (j) = i fa + 7 ^ + -Q dJ + v * > 



260) 



dt \Q/ Q dx g dy 



d \ g dw r 3w 



Under the circumstances of slow motions these also reduce to 



261) 



Thus we see that the three components of the vorticity are propagated 

 independently, each according to the equation for the conduction of 



