Ore — Stahility or Instahility of Motions of a Viscous Liquid. 127 



1 d^ V/da^ = (c/- 4 r-y {q" cosh 2q'a sin 2ra + r sinh 2q'a cos 2?u} 



+ 429^ {(q'^ - oq^r-) cosh 2^a sin 2ra + (3q'~r - r^) sinh 2q'a cos 2ra 



+ 4i2yq'r sin 4pa } , (25) 



-Yed^ V/da^ = (q'^ + r'^Y {(q'' - r") sinh 2q^a sin 2ra + 2qr cosh 2(fa cos 2?'a} 

 + 4^^ { (9'^- 65''^7'^+ r*) sinh 2<^a sin 2r6& +4$^r($^^-r'*) cosh 2(1 a cos 2ra. 

 + 82?-^Vcos42?a}. - (26) 



When a is zero, the first three differential coefficients vanish, and the fourtli 

 is positive. Substituting the values of q , r, given by (20) and (21), (26) gives 

 JgfZ* Vldcc^ = Q4:ifP{if + l^) sinh 2qa sin 2ra 



+ 64^^ (^^ + l^y(Sp' - ly- cosh 2^'a cos 2ra 



+ 32/ (/;^ + pf(3p' - IJ cos 4|m. (27) 



This cannot vanish for any value of 2ra less than 7r/3 ; since for such values 

 the second term exceeds the third even on replacing cos 4^a by - 1, and since 

 the first term is positive. Therefore, neither can F itself vanish, if 2ra < n/S. 

 Again, V may be written 



(6/ + 21-) sinh 2q'a sin 2ra - 2 (p^ + l^y (Sqf- - l^Y cosh 2q'a cos 2ra 



+ 2 (p- + Pf i?>if' - pf cos 4.qx(,, (28) 

 which, when sin 2Ta is positive, is algebraically greater than 



2 (p- + py (3^^ - l-y [ 3^ sinh 2q'a sin 2ra - cosh 25''a cos 2r« + cos 4^9a}. (29) 



Of the terms in brackets, when 2ra lies between 7r/3 and 7r/2, the first term is 

 greater than f sinh 2q'a ; the second is numerically less than -i- cosh 2qa ; and 

 thus the three are algebraically greater than f sinh 2q'a - ^ cosh 25''a - 1, and, 

 as q>r^2>, this is certainly positive. And, since q'>ryz, it is evident 

 that (29) cannot vanish if 2ra lies between 7r/2 and tt. Thus (18) has no 

 solution for which 2ra < tt. 



When 2ra > tt, sinh 2qa and cosh 2q'a each exceed 100 ; and accordingly 

 in (18) we may neglect the term involving cos4:2Ki, and may equate sinh2^'«. 

 and cosh 2q'a ; the equation thus sensibly becomes, making use of (28), 



tan 2ra = (p' + l~fi:?>f - rf {Zf + P)-\ (30) 



The simultaneous equations (21), (30i have, of course, an infinity of solutions; 

 there is one for which 2ra lies between tt and 47r/3 ; it may be shown that 

 there is only one; for, by the aid of (21), we may write (30) in the form 



r-i tan 2ra = {2p ^f^-P + f + V-^iof + P)'^ ; (31) 



as p increases beyond the value ll\/o, the right-hand member continually 



[17*] 



