106 Proceedings of the Uoyal Irish Academy. 



Art. 19. The Double Hoots of the Feriod-£!quatio7i. 



As for waves of sufficient length in the direction of flow, all the values of 

 2) are real, it follows that, if this wave-length be supposed at first large and 

 then to be gradually diminished, a value of j9 can become complex only by 

 the wave-length passing through a value such that two real values of p 

 become coincident. 



Now, if we write 



the period-equation in the notation of equations (6), (7) assumes the form 



0(FO^(r,)-(/>(F,)^(FO^O. (40) 



If 2) has the real negative value which makes 



Yi^ = Fj^ = a real negative quantity,* 



the functions (p{ Fj), ^(Fo) are identical ; and the same is true of 



Fr^^(F), F,-^^(F,), and also of xP^Y,), xp\Y,); 



accordingly, if this value of ^ just alluded to makes 4^{Y{), and therefore also 

 ipiYz) vanish, this value of p is a double root of the period-equation. (If such 

 a value of jp, however, makes xp ( Fj), ip ( Fj) vanish instead, it is only a single 

 root ; for, to be a double root, it would require to make either xp'(Y) or <^ ( F) 

 vanish; but no root of Jn{x) = can satisfy either J\i{p:) = or J_n{^) = 0-) 

 Thus, there are double roots ^j for certain values of /, 2^ and I being given by 

 the equations 



v,V-., = -3-i.0„. ^IK^JUo. (41) 



It may be proved, also, that these equations give the only double roots. 



The ecjuation 



r?M_?;S,^(F0^(F3)-^(F,)^(F0) =0, (42) 



which a double root must satisfy, when combined with (40), gives 



{0(^i)r-!0(^^)f (^3) - f (F,)^(F,)} = {c/>(F)J^-|,/,(F)f(FO - ./,'(FO^(FO]. 



(43) 



But, from the linear differential equation satisfied by ^, 1//, we have, for all 



values of the parameter, 



^ ( F) f ( F) - ,^'( Y)^p ( F) = constant : 



so that (43) is equivalent to 



[<1,{Y,)\'= (c^(F.)r-; l44) 



* For any such value i?' is represented by the point G (fig. 2, p. 108). 



