LIQUID SPHEROID OF FINITE ELLIPTICITY. 
203 
are positive 
we have 
for all values of n greater than unity." From our equations (49), (50) 
— (C) = ^3 {Vj = co7(47rpy i), 
SO that K 7 {i) is essentially negative and q=i{C) positive. 
In accordance with this, we shall now show that, if K/ [Q q.^ (1) be both positive, 
the roots of the period-equation for harmonic waves of degree n and rank s are all 
real, and we shall find their situations. 
In the first place, let us suppose s is different from zero. The period-ec[uation ( 66 ), 
as it stands, may be written 
F(k) = 0 , 
where 
NF {k) = siiF {4(7, (cot «) k (/c ~ 1) - (cot a)] {v) 
— (1 — K® siiF «)("-«- 1V2 gg(3 ot K,/ (cot a) (k — 1) [v), 
if we write for brevity 
N = ( 2 ?i) !/{ 2 ''n \ [n — s)]} 
We know that the roots of the equation 
DT 4 v) = 0 ..(70) 
are all real, and lie between -f- 1 and — 1 ; also they are separated by those of 
D^-iP 4 v)=: 0 . 
Let Ki, /C 3 , . . . K„_s be the values of k (taken in descending order of magnitude) corre¬ 
sponding to the roots of (70). These values of k all lie between + 1 and — 1 , also v 
decreases as k decreases. Moreover, if k is put in turn equal to k,, . . . the 
corresponding values of D’'^^P„(t') are alternately positive and negative. 
We are now in a position to trace the changes in F (k) as k decreases from + co 
to — 00, 
When K is greater than cosec a, v is imaginary. But F (/c) when written in the 
form of the left-hand side of ( 66 ) is obviously real; also when k — od the sign of F (k) 
is that of the coefficient of It is iherefove positive. 
When K passes through the value cosec a, v becomes infinite and then becomes real, 
but F (k) does not in general change sign. 
When k: = 1 , z/ = 1 , D^P,^^') is positive, and F (k) is negative. 
When K = Ki, F («:) is positive. 
When K = Kg, F (k) is negative. 
When K = Kg, F (k) is positive. 
and so on ; thus, when k = F {k) has the same sign as _ 1 , 
* We shall in fufcnre leave out the suffixes in and using e to denote the sizrface values, as these 
surface values alone occur in the remainder of our investigations. 
2 D 2 
