LIQUID SPHEROID OF FINITE ELLIPTICITY. 
205 
If n be even, the least positive and negative roots of (71) are and k,^i + i, also 
= — Kx,i- If the ratio K;, (Q : he positive, we find that the positive roots 
of the period-equation (62) are situated between the following values ;— 
00 , K^, K.2, . . K^n, 
while the negative ones which are equal and opjDosite to them are situated in the 
intervals between 
— oo, K,i, + 
there bemg no roots between and ki>i + 1 . 
When {Q vanishes, the roots are the 7i quantities 
^1’ 
none of which is equal to zero. If the ratio K„ {Q : q^iC) now become negative, the 
roots of (62) will at first continue to be real, being situated between the values 
Ki, K^, . . 0, . . •, K,i. 
This will be the case until we arrive at a value of ^ for which the period-equation 
has a pair of equal roots, each equal to zero. When this is so, we have 
(^) _-j. ^ rP„ (v) __ 1 
= n{n + l)> 
whence, 
or 
Pi (0 qi iO -Pn (0 (0 = 
When the ratio K„ {jQ) '• ~ Pi iC) or K,^ (^) : (^) becomes greater than 4/{^^ {n -f 1)}, 
two of the roots of the period-equation will become imaginary. 
In evei'y case there must be at least one jDositive root between each of the quantities 
^2) • • •} 
and corresponding negative roots, so that under no circumstances can equation (62) 
have more than one pair of imaginary roots. 
Numerical Solutions of the Period-Equations. 
17. For a spheroid of given eccentricity, a, and therefore are known. Now, the 
functions (C), tp (^) can be expanded in finite terms of { in exactly the same way 
