98 Proceedings of the Royal Irish Academy. 



of TTi ; and thus, if we make the further supposition, that the quantities 



^/3 / vX- + ]) 



are sufficiently large, equation (8), which expresses that 8 as given by 

 (3) should vanish for two different values of the parameter, takes the 

 approximate form 



2 , //3 ( v\" + 13 .Y( 4 2 ( /j3 ( v\~ + p 



3 I 

 or 



;/3 ' "Vl "3 171" -7^3- -'"I' ^'■'"' 



■,3| A 



where r is any integer, positive, or negative. 



If r is sufficiently large, whatever be the values of /, A, this equation 

 in p has one root such that the real part of vX^ + p, and a fortiori the 

 real part of p, is negative. (When the equation is rationalized, care must 

 be taken to distinguish between it and the equation which would be obtained 

 by connecting the two terms on the left-hand side by a plus instead of 

 by a minus sign.) In fact, as we have already supposed that a is small 

 compared with {v\~ + li)ll\i, the equation may be replaced by 



giving 



p = -v(X'+r'7r'l4a% (15) 



a value which is wholly real and negative. The suppositions made in arriving 

 at this approximate value of p, viz. : that (i^X" + 2^)/^(^ has its real part negative, 

 large compared with its imaginary part, and large compared with a, and that 



IB fvX^ + p 



are sufficiently large, are accordingly justified, provided r is sufficiently 

 large. And as r may be any integer if large enough, it thus appears that 

 the approximate form of the period-equation has an infinity of roots. 



Moreover, from the value foimd for jh it appears that by taking r large 

 enough, the accurate form (8) of the period-equation may be represented as 

 closely as we please by the approximate form (14), so that the actual period- 

 equation must have an infinity of roots. 



