134 Proceedings of the Royal Irish Academy. 



which reduces to 



9 3 32.961.[9 512.6223.1 9 512.802.f 9 



(72) 

 This lias a root in the neighbourhood of a^- = 44. The minimum vahie 

 of - li-ui is bj no means sharply defined; the values \:o, 4--i, 4-5 substituted 

 in (70) give - U'cu^ = 7591. 7565, 7576 respectively. These all giA'e 



Cpcr/^ = - il'a' =87. . (73) 



In (70), however, the terms invoMng the fourth and higher powers of h 

 have been neglected. If we substitute the values which have been found 

 for Jc and oi in the two terms involving /j* in (69) the former would raise 

 the value of - Jc'-cii'^ by about 1 per cent., and the latter by about one-fourth 

 as much. "\Te would presumably make proper allowance for all the terms 

 neglected* if we increase the value found for - Jc-cii^ hj 2 per cent., or that 

 of - ikcii^ by 1 per cent.. Thus we would obtain the criterion 



BUpl^ = -^Cpa'jSiu = 117. . (74) 



Aet. 34. Floe: through a Circular Pipe. Sharpe's Eesult. 



The case also of flow through a circular pipe has been discussed by 

 Sharpe.f Taking the z axis in the direction of flow, he selected an initial 

 distur'bance in which 



2au = Z^7r?'(sinp -i- sin op) Bmirhj'la - IB-r (sin p -r 2"^ sin -ij)) cos Trhj2a, 

 2au- = A[4:a(&mp + sin 3jj) + 7rr(cosj; + 3 cos 02))] cos 7r/«/2« 



+ B\4:a (sin 2^:* + 2~^ sin 4/;) + tt/' (2 cos '2p + 2 cos 4j:>) j sin ~Iz/2a, (75) 



where u is measured radially, c: in the direction of flow, the radius is a, and 

 p) denotes 7rr/2«. On investigating the values of B/A and of /, which lead to 

 the greatest possible value of p for which the disturbance could, be stationary, 

 he arrived at the equation 



DpWJiii = 2>ypTr!fA = 470, (76) 



71^ being the mean velocity in the steady motion. I believe, however, that 

 his work contains a numerical error J which sensibly aftects the result ; and 

 that if this were corrected, the number 470 would be reduced to about 363. 



* It appears that we may safely uegiect terms in ■vvhicii occur k^ or higher powers ; for the left- 

 hand member of (62) forms part of the left-hand member of (69) ; as far as can be judged, the terra 

 involving l-^ in the former is the most important term inTolving it in the latter;- and substitution of 

 the numbers just found sho^rs the value of this term to be about 1/20000. 

 t Loo. cit. - - - 



i A coefficient of £-1- in a certain equation -which Sharpe gives as G"67 should. I think, be 



(ttI - 27577- 24 -f 1312 27 1 16 or 2-057. 



