130 Proceedings of the Royal Irish Academy. 



Substituting 2ar = 57r/6, tt we ol3tain l-lxf" = "73, 'lo, respectively. The 

 former value suljstitutecl in (44) gives 2ra to be less than it by the circular 

 measure of 20° 54' ; and the latter 20° 42' ; we therefore see that the correct 

 value of I^Ijp^ is nearly '736, and that of the angle in question 20° 50'; 

 thus 2ar = 2-778, and finally 



Bpay,x = 26-36 or BpD^fji % 105-5. (47) 



If, again, we were to take as boundary-conditions 



dv/dy = 0, dHjdy''- = 0, 



we should obtain equation (13) over again, and the same criterion as in (38). 



Aet. 32. A Stream hetween fixed Parcdlel Planes. Besidts of Reynolds a.nd 



of Shaiye. 



The case of flow between fixed parallel planes was the only one to which 

 Eeynolds himself applied his method so as to obtain a numerical result.* 

 Noting that if the disturbance is expressed as a trigonometrical function 

 of y, the higher harmonics would, on the whole, make for increased stability, 

 he chose as the type to be investigated one in which 



% = A (cos 23 + 3cos3p)cos-fe/2a + i?(2cos2jj + 2cos4^j)sin7rZ5c/2a, (48) 

 V = /^(sinjj + sin3jj)sin7r/«/2a. - lB(siR22) + 2"^sin4j;)cos7rfe/2«) (^^9) 



where p = 7ry/2a. - The values of I and of B/A were then so determined 

 that the value of /x obtained by equating to zero the rate of increase of 

 the energy of disturbance should be greatest possible, and the result he 

 obtained for the critical equation was 



BUplfx = 017, (50) 



where I) = 2a, the distance between the planes, and U is the mean velocity. 

 This case has also been discussed by Sharpe ;t he chose as the type 

 of disturbance that in which, in the same notation, 



u = A(sm2} + sin3p)cos7rfe/2« + B(2siR223 + 4sin4^j>) &iii.7rlx(2a, (51) 



-y = - lA(cos2J + 3"^cos3j9)sin7rfe/2a + lB{cos22) + cos4:2J)cosTrlcc/2a, (52; 



and obtained a lower value for the critical velocity, his equation being 



BUplfx = 167. (53) 



* Loc. cit., p. 75, ante. 



t " On the Stability of the Motiou of a Viscous Liquid " : Trans. Amer. Math. Soc, vul. vi. 

 No. 4, October, 1905. 



