90 Proceedings of the Royal Irish Academy. 



corresponding value of u. It is 



2hi sinh lb _ Exp [~ vt {P + m~ - lm(5t + r^jd'tyS)] 

 [l^m^B " ~" Z" + {in - l(5ty- 



X { -{m-l^t) sinh Ih sin[&+ {m-l(5t) ■i/]+l cosh l(h~y) cos Ix-l cosh ly cos[fe+ {m-l^t) H] j 



Exjp [- vt{l'' + m- + Imjit + PjiH'l^)] 



r- + (m + i(5ty 



X { («i.+//30sinh/6sin[&-(??^+Zj3;^)2/]+Zcosh/(&-2/)cosfe-Zcosh/2/cos[/£c-(7?n-//3!^) h]]. 



(30) 

 It is seen that these expressions differ from those obtained when viscosity 

 is ignored (Part L, equations (28), p. 26; (38), p. 28;) only by the presence 

 of the exponential multipliers, and become identical with them if v is equated 

 to zero. There thus appears to be no necessity for the suggestion thrown out 

 by Lord Eayleigh that, in these questions of stability, investigations in which 

 viscosity is altogether ignored may possibly be inapplicable to the limiting 

 case of a viscous fluid when the viscosity is supposed infinitely small* 



Art. 11. For suitable Values of Constants in First Modification the Disturbance 

 will Increase greatly. Suiistitution of a mcmerical Vahce suggested 

 hy Experiment. 



Taking then the values of u, v given by (29), (30), they are derivable from 

 a stream function, ip, given by 



2/^ sinh _| ^ ^^ ^ ^^^ _ ^ 12^2 fm 



(r + m^) B ^ ^ / I ' J 



sinhZ6 cos[& + (m - l(5t)y] - sinhl {b - y) cos Ix - sinh/?/ gos[Ix + {m - l(5t)b] 



l' + {m- l(5ty 



- another term derivable by changing the sign of m. (31) 



Here 



- 2/V-^ 



Exp [- vt{l' + m- - ImjSt + l'(5Hy3] . cos[fe + (m - li3t)y] 



{P+m")B 



- another term derivable by changing the sign of m. (32) 



If T be the average energy of the relative motion per unit length of stream, 

 4:Tn/l = - ('" I xl^V'ipdxdy. (33) 



Jo Jo 



Making use of this, on performing the integrations, and comparing the value 



* "On the Qaestioa of the Stability of the Flow of Fluids," Phil. Mag., xxxiv., p. 61, p. 67, 

 1892; Scientific Papers, iii., p. 577, p. 582. 



