Orr- — Stabilitf/ or Instahility of Motions of a Viscous Liquid. 107 



and thus the equations to be satisfied in case of a double root are either 



0(ZO-0(F,X and ^(FO = ^(F,), (45) 



0(FO = -0(r.), and ^(FO = -^(F,). (46) 



The former alternative is equivalent to the statement that ^(Fi),^(Fi) should 

 both be purely real ; the latter, that they should both be purely imaginary. 

 In either case, there would exist some equation of the type 



^(FO + C'^IFO^O, (47) 



in which is some real quantity, except either (p or t// vanishes (for both 

 Fi and F2). Of the two exceptional cases, that in which 



^(F0=^(F,) = O, 0(F,) = 0(F), (48) 



is the one already referred to ; for, as a Bessel function* can vanish only for 

 real values of the argument, the former pair of these equations requires 

 F]^ and Fj^ to be real, negative, and therefore, by (39), equal, quantities. 

 The second exceptional case, i.e. 



,^(FO = 0(F.) = O, ^(FO=^(F,) (49) 



is impossible, for the former pair of equations again requires that Y^^ and Yn} 

 should be real negative equal quantities. Then, since Fi cannot be equal to Fj, 

 the second pair would imply that 1// ( Fi) and ^ ( F2) should both vanish ; this 

 would recover the former exceptional case, though it is impossible that 0, ;// 

 should vanish together. Thus we are driven back to equation (47). But this 

 cannot be satisfied by a complex value of FI We may rest this last statement 

 on the general theorem that, if n lies between + 1, any expression of the form 



where C is a real quantity, and every power of x has its principal value, can 

 vanish for, at most, only one value of x, and this a real positive one.f Or it 

 may be established independently as follows : Denote by x[Y) the left-hand 

 member of (47) with Fi replaced by Y;X and suppose, if possible, it vanishes 

 for F, and F2, complementary complex values ; we evidently have 



(Px (« Y,)lcW = a Y^x (a ^0 , 



f?:x(«^2)A^«' = aF«x(aF); 

 from which we deduce 



X (« Y,)d' x(« ^^) A^«^" - X (« ^^)d\ (« ^1) A^«' ={Y^- F.^) ax (a FO X (a Y,) ; 



* Of Older greater than — 1, as here. 



t Unless n = |, in which case it may be a negative one. 



t By r is denoted {lfi/v)s{- v\~ - j) - Wyi)ll^ as in (6). 



