Orr — Stability or Instahility of Motions of a Viscous Liquid. 83 



The value of v satisfying (2), (3), (16), (17) is obtained by first finding 

 an imaginary type solution* Assume 



V = e'>* + '^+"=)F (18) 



^^^i[^t^lx^nz)^^ (19) 



Equation (2) then becomes 



:^J,.n'.'^-*'mS. (20) 



This may be solved by series proceeding in ascending powers of 



V' + -/i^ + 'i(aj + l^y)\v 



which are seen to be essentially convergent for all values. The form of 

 8 having thus been found, the solution of (2) can be expressed by using 

 integral forms, and it involves four arbitrary constants ; by the aid of these 

 arbitrary constants, any prescribed values can be given to v and to dvlcly 

 for y = and y ^t). Thus a real value of v satisfying (2), (3), (16), (17) may 

 be obtained, 



ISTow, the V solution, expressed by (13), comes essentially to nothing 

 asymptotically as time advances. Hence, Lord Kelvin states, the i) of 

 (2), (3), (7), (8), which rises gradually from zero at ;J = 0, comes asymptotically 

 to zero again. He concludes that the steady motion is stable. 



Art. 3. His Solution of the Second Prohlcni and, its 'iiiodificatio7i to 

 suit the First ProUem. 



In the second paper, which, as stated above, deals with the case in which 

 the steady velocity is expressed by a quadratic function of y, Lord Kelvin 

 writes as in (18), above, 



,y ^ Qi{mt^lx-vnz) Y 



and obtains the differential equation satisfied by V, which is of the fourth 

 order. He shows how four independent solutions of it may be obtained in 

 the form of series in ascending powers of y, convergent for all values of y, 

 unless V be zero. The rest of his discussion is by no means full ; I trust I do 

 not misinterpret it in the following statements. He appears to indicate that 

 by means of the four arbitrary constants which occur in the value of V, any 

 values desired can be assigned to V and to d V/dy for y = and y = h, and 

 that by integration or summation with respect to to, I, n, one can thus obtain 

 the motion produced in the fluid by giving the plane boundaries y = 0, y = b, 



* At this stage of Lord Kelviu's work, in his equation (49), there occurs an error which is noted 

 in an " erratum " prefixed to the bound volume of the Phil. Mag. 



