128 Proceedings of the Royal Irish Academy. 



decreases, while the left-hand member continually increases, for r, given by 

 (21), continually increases. And it is this solution which we require ; for 

 (21), (22) show that, / being given, the smallest value of r corresponds to the 

 largest value of ^ for which the disturbance could possibly increase. 



We finally wish to obtain the greatest value which the value of ^ so found 

 can be made to assume by varying I. A stationary /t is a maximum ^, for /x 

 has no minimum ; as I increases indefinitely, r remains finite, ra being < 47r/3, 

 and^:), satisfying (21), tends to equality with Ij^^o, so that fx given by (22) 

 diminishes indefinitely. The differentiation of (22) gives us for a stationary fi 



fdlldp = {Sf + 2P) I. (32) 



By differentiating (30), making use of this, we obtain 



ap' {Sp- + 2P) {Sjf - Pf dr/dj) = - 21' [f + pf ; (33) 



and in a similar manner from (21), 



frdrldp = 2jj {f + Pf - {f + P) (if + 2P). (34) 



Combining (33) and (34), there results 



a {2>f + 2r-)(3^/- - Pf { [f + 2P) [p^ + Pf - 2iJ [f + /')}= 2Pr, (35) 



and this, (21), and (30) are equations determining /, jj, r. From (21) and (35) 

 we obtain 



2ra (3^^ + 2^^ [/ + 2/^-22? [f + P)^ = 4.P [2-p - {jr + Pf]Cif - /T*. (36) 



If 2ra were Vtt/G, the value of P/p^ which would satisfy this would be '93 ; 

 while, if 2ra were tt, it would be '94. It will be seen that the former 

 supposition is very nearly correct; taking then the former value of P/j)'^, 

 substitution in (30) shows that 2ra is the circular measure of 206° 57' (the 

 latter would give about 3' less), i.e. 2ra = 3' 61. From (21) there is next 

 obtained l/r = 1-05 (and < 1"06), giving la = 1-89. Then (22) gives 



Bp/{8r'f.i) = ft' { 2p - /j/- + P ; -' = 1-698 (and < 1-699). (37) 



Thus, il D = 2a, the distance between the bounding planes, there finaUy results 



Bpa'l^ = 44-3 or Bpl?:,x = 177. (38) 



This result has been obtained on the supposition that the initial disturbance 

 has a definite, but imdetermined, wave-length; but as the different wave-lengths 

 contribute to the rate of increase of the energy of disturbance terms which 

 are simply additive, this restriction may be removed, provided the proper 

 end-conditions are satisfied, and for tliis it is sufficient that on every stream- 

 line the end- values of the velocities and of the alteration in pressure should 

 be the same. 



