( 695 ) 
The expressions [pq] all contain m, as a factor. Thus, in order 
to derive the term independent of m, in the development of @, we 
take all those expressions = 0. The determinant then becomes divisible 
by 0‘, and is reduced to its first four columns and rows. Four of 
the eight roots of our equation thus appear to be divisible by Ym,. 
The first terms of the other four are the roots of the equation: 
—o 0 s 0 
0 -—e 0 s' 
= 0, 
An A. cage 0 
B As 0 —oe 
or 
o* Tik, (ar 8 zi Aln s) o Sin (A,, Ay. a AG ve) ss = 0 ic 2 2 (7) 
where we have put, for brevity: 
C35 MCR = st. 
The solution can only be stable, if the equation (7) has two real 
and negative roots. Now A,, and A,, are negative, and A,, 4,,—A’,, 
is positive. The necessary and sufficient condition that the equation 
(7) shall have two real and negative roots is therefore, that both 
s and s’ are positwe. Now we have 
sat l Acme Beale +a | 
a, : : 
ER al) 
pier Ye, ' + ( 
s' = —{ de, cos a — Be, cos (a +2) | 
a, l 
For the six possible combinations we find the signs of s and s’ 
as given below 
| : 
a a | 8 | bet 8! | s | 8 | 
| | sr 
(6) | 0° 180° 0°) 1809 ie ae 
(16) | 480 | 180 | 480 | 180 | — | —] unstable. 
IR A EN (130 | oO | + 
mo |19 | 18: | 0 |+/0 
(12) [180 | o | 480 | 480 | O | — | unstable. 
(13) | 180 | 480 | O 0 — | O | unstable. 
+7 
Proceedings Royal Acad, Amsterdam, Vol. XI. 
