384 Proceedings of Royal Society of Edinburgh. [sess. 
The partial eliminants are formed as before by excluding 
successively sin <£, cos 2 <£, and 1 ; sin 2 <£, cos <f>, and 1 ; sin <£, cos <£, 
and l 2 ; as thus, 
[sin <£] 
[cos 2 (f\ 
[i] 
(1) 
2B cos <£ + C sin <£ 
A 
a COS </> ~ X 2 
(2) 
— 2B cos cf> + A sin <f> + /2 
C 
- y 2 
= 0 
(“) 
(4) 1 
sin <f> 
1 
-l 
[sin 2 </>] 
[cos </>] 
[i] 
(1) 
C 
2B sin cf> + A cos <f> + a 
- cc 2 
(2) 
A 
- 2B sin <f> + C cos 
/3 sin $> — y 2 
= 0 
(») 
(4) 1 
1 
cos <f> 
- 1 
[sin (fl 
[cos </>] 
fi"] 
0) 
B cos <£ + C sin <£ 
B sin <£ + A cos <j> + a 
- x 2 
(2) 
— B cos + A sin <f> + f3 
- B sin <£ + C cos cf> 
- y 2 
= 0 
<«) 
(4) 
sin cf> 
cos 
-l 
The partial eliminants ( = 0) are, 
(a) (a J3 - 2 By) cos <J> + ( A - C)y sin <£ + A - x 2 ) + (A - C)a cos </> sin <£ - 2Ba cos 2 ^>. 
(&) (A - C)y cos <£ + (a p + 2By) sin <f> + a(A - y 2 ) + (A - C)/5 cos <f> sin <j> + 2B/3 sin 2 4>. 
(c) {y3(A - x 2 ) - Ba} cos <f> + {a(A - y 2 ) + B /?} sin <£ + aj3 + (A - C)y cos sin <j> 
+ By sin 2 <f> — By cos 2 <f>. 
To make these expressions symmetrical a slight transformation 
is requisite in the last terms of (a) and (b). 
In (a) substitute Ba sin 2 <£ - Ba cos 2 < p - Ba for ( — 2Ba cos 2 <£). 
In (b) substitute B /3 sin 2 cj> - B/3 cos 2 <J> + B J3 for ( + 2B/3 sin 2 <£). 
The three equations ( = 0) are then, 
(a) (a f} - 2By) cos + C)y sin <#>+ {/?(A - #) - Ba} + a j {( ^ " / C) C ° S * ™ f } 
(A) (A - C)y cos $ + (a/3 + 2By) sin 0 + {a(A -if) + B/3} + /3 j _ C ) cos ^ sin ^ 1 
( +B(sin 2 0 -cos 2 ^)} j . 
(c) {/S( A - z 2 ) - Ba} cos * + { a( A - 2/ 2 ) + B/3} sin 0 + OJ 8 + y j {( ^ 7 , C > C ° S * S “ * j 
(. + B(sin 2 </> — cos 2 </>)} J • 
The same expressions can be got from the equations of addition 
and subtraction of (1) and (2) along with (4). 
