118 Proceedings of the Royal Irish Academy. 



The boundary-conditions thus lead to the period-equation 



S.e'^ydy [ S^e-'^ydy - ['^ S.e-'^ydy \ S^e^^dy = 0, 



(70) 



where S^, So are any two independent solutions of (1). 



A laborious development of this equation in ascending powers of jj' threw 

 little light on the nature of the roots ; every term in the equation appears to 

 have the same sign, however. 



On the supposition, justified to some extent by results, that for all the 

 roots the quantities which occur as variables in the Bessel functions in S are 

 large, an equation approximately equivalent to this may be obtained. As 

 approximate forms of S are (-_// - l^yiyi. e-", where 



u = 



1(5 )' 



ai) 



it might appear that we would be justified in using these exponential forms 

 in the integrands, and replacing, for example, 



by 



(-y - l^ijiyie''^^ydy 



(- ]p' - l^yi)-k 6«+^->'/(X + duldy) 



Irrespective of the delicate considerations of the discontinuity in the forms of 

 the approximate expressions for the Bessel functions, this procedure would 

 not, however, be prima faxie justifiable unless it were possible, regarding 

 iy as a complex quantity, to connect the limits of integration by a path 

 along which the real part of u + \y continuously increased, or continuously 

 decreased, which is not always possible. I therefore considered more fully 

 the functions le-^^Sdy; but the approximate form finally obtained for the 

 period-equation proved so intractable that it does not appear justifiable to 

 go into details. In the region in which the roots appear to actually lie, yiz., 

 one in which ^f has its real part negative, and its imaginary part between 

 the limits + l^ai, the form is 



(-//(^i3) + «^H 



Exp (Xa + Ui) 



-2-fe)'^ 



-X-i[[-p'+l^ai)lv} 



\+i{{-p'^l^ai)lvf 

 vXH - 2i\p'i 



31(5 



i Exp (Xa - «i) 

 X-i((-p'+l(5ai)lv)\ 



Exp (- Xa + Uo 



-(-//(//3)-«iH 



Exp[-Xa-Ui] 



X + i{{-2y-l(5ai)/v)i 



-X-i{{-2y-l(5ai)/v)i 



