158 ME. S. S. HOUGH ON THE APPLICATION OF HAEMONIC 



For the particular case s = 0, the biquadratic to which the equation l = is 

 equivalent reduces to a quadratic, the two roots which remain finite being of equal 

 magnitude and opposite sign. This special case has been examined in Part L, and it 

 will be seen on reference to the tables there given ( 7-8), that the roots of the 

 equation L,, = give a very good approximation to the roots of the period-equation 

 except in the case of the earlier roots when kg/Auto 2 is small. In the present paper 

 I have examined in some detail the special cases corresponding to the values 1 and 2 

 for s, and the approximation is found to be equally rapid, as will be seen from the 

 tables given hereafter. Consequently, in these cases at least, all except the two or 

 three smallest roots will be obtained with adequate accuracy by finding the roots 

 which lie in the stated intervals of the equations L' = with different integral 

 values of n. 



The roots so found will not however form the complete series of roots of the 

 period-equation. We may in fact anticipate that the remaining roots of the equation 

 L'j = will also approximate to roots of the period-equation. To obtain a better 

 approximation of the roots of this class, it will however be preferable to make use 

 of the period-equation in the form (57). As a first approximation we omit the 

 continued fractions and obtain 



, , 2o>.s 

 N" = or n (n + 1 ) = 0. 



A, 



This method of approximation will be valid if when X/w = 2s/n(n + 1) the two 

 continued fractions involved in (57) are small in comparison with n (n + 1). But it 

 may readily be verified that with large values of n these continued fractions become 

 comparable with a) 2 a?/hg, and therefore the desired condition will certainly be 

 satisfied when n is sufficiently large. 



A better approximation may be obtained by representing the continued fractions 

 by their first converger) ts. We thus obtain as the approximate form of the period- 

 equation for the determination of the root which lies near - 



* M I an . 



N' - 



n (n + 1) 

 a', a' 



or 



(n - I) 2 (n 4- I) 3 (n - s) (n + s) 



]:(:: | n _ 



' ' 



(n n 2ws l 

 Tj 



(n + 2)3 (n - s + 1) (n + s + 1) 

 (2n + 1) (2*1+3) 



