380 Proceedings of Royal Society of Edinburgh. [sess. 
The required determinant, of the 4th order, is formed from (3), 
together with the three dialytic eliminants got by eliminating 
successively cos 2 #, sin 6 and unity ; sin 2 #, cos # and unity ; and 
cos #, sin #, (l) 2 , between (1), (2), and (4). The three partial 
eliminants are — 
[sin #] 
[cos 2 #] 
[i] 
(1) 
C sin # 
A 
a COS # — X 2 
(2) 
A sin # + J3 
C 
-f 
0) 
sin # 
1 
-1 
= a (3 cos # + (A — C)y sin # + /?( A — x 2 ) + ( A - C)a cos # sin # = 0. (a) 
[sin 2 #] 
[cos #] 
[i] 
(1) 
C A 
l cos 6 + a 
— X 1 
(2) 
A 
C cos # 
/S sin # - 
- y 2 
= 0 
(*) 
1 
cos # 
-1 
C)y cos 0 + a/3 sin # + 
a(A-y*j + (A- 
- C)/3 cos # sin 
# = 0. 
[sin #] 
[cos #] 
[13 
(i) 
C sin # 
A cos # + a 
— X 2 
(2) 
A sin # + j3 
C cos # 
-f 
= 0 
(4) 
sin # 
cos # 
-l 
= /?( A — x 2 ) cos # + a(A - y 2 ) sin # + a/3 + ( A - C)y cos # sin # = 0. (c) 
The final eliminant is, 
[cos #] 
[sin #] 
[1] 
(a) 
a/? 
(A-C)y 
y8(A - a? 2 ) 
W 
(A-C)y 
a/3 
a(A - «/ 2 ) 
(o) 
y8(A-* 8 ) 
a(A - ?/ 2 ) 
a/5 
(3) 
a 
/3 
7 
[(A - C) cos # sin #] 
a 
P 
7 
= 0 • (Ac) 
If we write this bordered symmetrical determinant in the 
generalized form, 
a/? 
l 
m 
Z 
a/3 
n 
m 
n 
a [3 
a 
/3 
7 
a 
7 
