Ore — Stahility or Instability of Motions of a Viscous Liquid. 133 



where, in determining a, y is equated to a. Denoting this boundary-value of « 

 by ai, this equation, after division by oi, becomes 



+ (2048 + 2816/j-) r^ + (8192 + 128.168^^) -^ 



■ I io I lo 



+ (32768 + 1474o6A;- + 15360^'0 pf^ "*" ' ' ' 



+ (131072 + 32^912/c^ + 276480^^0 "^ + • • • = (69) 



L— 

 In verification of the somewhat lengthy numerical work involved in calcu- 

 lating the coefficients in (69), I obtained it as far as the terms involving Tr in 

 another way, using solutions of (58) in the form of series which proceed in 

 ascending powers of }:, the coefficient of each power being a function of a. 

 This method did not appear to have much advantage over the other. The 

 portion of the left-hand member of (69) which is independent of h is 



(2ai + sinh 2cti)/4ai. 



We have now, regarding /, and therefore oi, as given, to solve (69), choosing 

 the highest root in /.t, and therefore the lowest value of Ji. Then / has to be 

 chosen, so that this value of }x is the greatest possible, i.e. the lowest value of 

 - iUa^ is to be made a minimum. The lowest value of - Jc'ai is, approxi- 

 mately, 



,. , . , 2or 8ai* 32a,^ 128ai« 512ai^« 2048ai^- 



^^ /«;- lOai^ SSai*^ 672ai« 4608011^° 21504aii- ) 



(70) 

 in which terms involving k^ have been neglected. Making this stationary, 

 we obtain the equation 



-J- 



3 



(«i^ lOai^ SSai'' 672«i« 4608aii° 29184a,i' 

 — ' — + + + + + + . . . 



■ ([_9 |_n [13 |15_ [17 [19 



2 2.8oi^ 3.32ni^ 4.128ai« 5.512ai* 6.2048a/* 



^ [!" TT " TT^ "TF^ TiT ^ "TlT" '■" 



_ (^ 2J0^ 3.88a^ 4.672ai° 5.4608ar^ 6.2918 4a,'° ^ 



^ iii ' TlT "" li!" ■" LL^ "^ (11 ^ II? '■■'■) 



(71) 



R.I. A. PROC, VOL. XXVII., SECT. A. [18] 



