Orr — Stahility or Tnstabilitij of Motions of a Viscous Liquid. 109 



that, when the period-equation is written in the form (23), we may take in 

 the notation of (26), A = 5/72, B=5 y 2/72. We shaU not be using the 

 approximations in any case in which the vahie of | ^Cl | or | Wj | at C is less than 

 oTr/4: ; consequently, at any point between C and 0, the value of \n\ exceeds 



(v/3/2)i . 37r/4 or 1-8989, 



and thus the fractional error in e"i or e"3 is less than 1/27, and that in e-"i or 



e'"2 less than v/2/27. Thus, if the period-equation be brought to the form 



- ^(e^-^ - r"2) + 1 % (50) 



by dividing across Ijy the factor which will make the third term rigorously 

 accurate, the fractional error in e-"i or e-"2 is less than 



and therefore less than 1/10. Thus the correct left-hand member lies between 



e^^(2 sin 2Q ± 1/5) + 1. 



Let us suppose that at C, Ui = U2 = niri -\- 7^^/4, where n is unity, or any 

 higher integer. At C the left-hand member lies between the limits 



2 sin 7r/2 ± 1/5 + 1, 



and is therefore positive. As '/ travels from C towards (9, the factor 

 2 sin 2$ ± 1/5 remains positive, certainly until 2Q decreases by tt/S, at which 

 stage 2P has decreased algebraically by more than 7r/3, (for it may easily 

 be seen by differentiating {-p' + ai)i that its real part decreases algebraically 

 as y moves towards (9 at a rate which, measured absolutely, is greater 

 than the rate of decrease of its imaginary part), and hence c^^<e~'^l3 <e'^ ; 

 everywhere between this pomt and 0, e^^(2 sm2Q ±1/5) is numerically 

 less than (2-i-) e~^, and thus the left-hand member is positive. Under these 

 circumstances, then, there is no root of the period-equation for which j^' li^s 

 between and 0. 



Let us next suppose that, at C, Ui = iQ = mri - ■ni/4:, n being unity or any 

 higher integer. At C the left-hand member of (50) lies between the limits 



- 2 sin 7r/2 ± 1/5 + 1, 



and is therefore negative. Again, at the left-hand member lies between 



the limits 



c^^(2sin2§±l/5) + l, 



where P is negative and numerically greater than (1'9)/y/'2, this being its 

 value in the case ?i = 1 ; from this it is clear that the left-hand member is 

 essentially positive. Thus, under these circumstances, there must be some 

 odd number of roots for which |/ lies between C and 0. 



R.I.A. PROC, VOL. XXVII., SECT. A. [15] 



