[1 
ON A THEOREM IN THE GEOMETRY OF POSITION. 
3 
i] 
ion to 
anishes, 
;o zero, 
situated 
Detween 
which is easily expanded, though from the mere number of terms the process is some 
what long. 
Precisely the same investigation is applicable to the case of four points in a 
plane, or three points in a straight line. Thus the former gives 
0, 
2 
12, 
2 
13, 
2 
14, 
1 
2 
21, 
o, 
2 
23, 
2 
24, 
1 
2 
31, 
2 
32, 
o, 
2 
34, 
1 
2 
41, 
2 
42, 
2 
43, 
o, 
1 
1, 
1, 
1, 
1, 
0 
The latter gives 
0, 
2 
2 
12, 
2 
13, 
2 
1 
= 0; 
21, 
2 
0, 
2 
23, 
1 
31, 
32, 
0, 
1 
1, 
1, 
1, 
0 
or expanding, 
4 4 4 2 2 2 2 2 2 
12 + 13 + 23 - 2 . 12 13 - 2 . 13 23 - 2 . 12 23 = 0; 
which may be derived immediately from the equation 
± 12 ± 13 = ± 23, 
and is the simplest form under which this equation, cleared of the ambiguous signs, 
can be put. 
(The above result may be deduced so elegantly from the general theory of elimi 
nation, that notwithstanding its simplicity it is perhaps worth mentioning.) 
Let 
X o - X u, = 
X n~ X l 
II 
3+ 
1 
s 
,/=75 
then 
12 =7 2 , 23= a 2 , 
2 
31 = ß 
2 , and 
a + 
/3 + 7 = °; 
from which a, 
ß, 7 
are to be eliminated. 
Multiplying 
the last equation 
a/3, and reducing by 
the three first, 
0 . a +1?. 
/3 + 31* 
7 + 
a/37 
= 0, 
12 2 . a + 0 . 
/3 + 23 2 . 
7 + 
aßry 
= 0, 
3Ï 2 . a + 23 2 . 
/3+ 0. 
7 + 
ctßy 
= 0, 
a + 
ß+ 
o 
+ 
c" 
aßy 
= 0; 
1—2