80 
ANALYTICAL RESEARCHES CONNECTED WITH 
[114 
( 2fgh 
\J (f-g+h) 
— f+ g + h + 2 J-- 
+ g 2 — h 2 
x 
+ 
/__2fgh_ 
[j (f-g + h) 
+ f — g+h — 2J 
+ 
— 2fgh 
./(f-g + h) 
+ f+g —h + 2/ .? + & + & 
2gh 
h 
i - 
j 
+ 2 \J 1 ~^j\J 1 ~^j\] 1 — ~ V w== ® 
( ~ 
U(f+g-h) 
— f + g + h + 2J 
f 2 - g 2 + h 2 
2fh 
a: 
+ 
+ 
/ - 2f g h 
U(f+g-h) 
/ 2f g h 
U (f + g - h) 
+ f +g - h -2/ 
+ 2 1 — 1 — j y ^ — pw = 0, 
values which might be somewhat simplified by writing £, 77, £ w instead of 
h , 
-y V —pw; 
and it may be also remarked, that the coefficients as well of these formulae as of those 
which follow may be elegantly expressed in terms of the parts of a triangle having 
f, g, h for its sides. 
The equations of the separators are found by taking the differences two and two 
of the equations of the resultors (this requires to be verified a posteriori)', thus sub 
tracting the third equation from the second the result contains a constant factor, 
equivalent to 
J ( f2 -(g-h) 2 )gh 
{4f 2 g 2 h 2 - J(i 2 - (g - h) 2 ) ((g+ h) 2 - f 2 )}, 
i_ [4f2 ff 2 h 2 _ ja (2 1 4f 2 g 2 h 2 \\ JK 2 
/(f 2 -(g-h)*)ghV s J \ K + j, )) 0 (P-(g_h) 2 )gh- 
Rejecting the factor in question and forming the analogous two equations, the equa 
tions of the separators are 
tA-Lfi 
f VMV 
X + 
X — 
X — 
h 
g_ 
~ f g_ 
g V23 
JL 
>/28 
y — 
y + 
'1