in certain Problems of Viscous Fluid Motion. 625 



Substituting in (7), we have 



r/£ y crA J X 2 + £(1 + &) 



- «r S X» + *(l + ife)' ••••••• W) 



the terms independent o£ t cancelling out on summation. 

 The velocity at any time is given by 



<n/ ^{X 2 + &(! + £)}' ' .. ' 



It can be verified by summation that the limiting steady 

 velocity has the value gtrkfe/j,. The fluid velocity at any 

 point can be obtained by substituting V from (21) in (2) 

 and reducing the expressions, but it is, of course, simpler to 

 insert suitable functions of x directly in (21) ; we obtain 



gah ( x\ _ Igph^s miXil-vlhtte-'W** '^ 

 2fju\ hJ av *\ 2 {\ 2 + k(l + k)}shr\' ^ ' 



In this particular problem the result can also be obtained 

 from the differential equation together with the boundary 

 and initial conditions, by assuming the existence of a 

 limiting steady state. In the preceding analysis the 

 existence of a final steady state is associated with the occur- 

 rence of zero as one of the exponents in the nucleus of the 

 integral equation (4). 



5. It is interesting to deduce the motion in an infinite 

 fluid from these results. In solving this case directly, 

 Rayleigh obtains the equation of motion as 



Applying Abel's theorem, this is reduced to an ordinary 

 differential equation whose solution is given as 



,1 00 



4t7Ti[ipV/ ga = 4:pvtf — 7ria-i 2ae 4 P 2vt<T - 1 e~ u2 du. (21) 



«. 2pv-t s /<r 



We obtain (23) from (I) by giving the nucleus its limiting- 

 value, since 



Lim (2/a/o7i) 2 e -" 2 *M*-T)/A2 _ (2pv^jair) \ e-^- r) dot, 



A">00 —00 ' — 00 



