j 7 m BRSM9E 
1 j- g > , i y x0.^ J' /1 
i] 
ON A THEOREM IN THE GEOMETRY OF POSITION. 
[From the Cambridge Mathematical Journal, vol. II. (1841), pp. 267—271.] 
We propose to apply the following (new ?) theorem to the solution of two problems in 
Analytical Geometry. 
Let the symbols 
a , /3 , y 
a', /3', 7 
0", 7" 
denote the quantities 
a, aft — cl ¡3, a(3 y" — o(3"y 4- CLf3'y — o'¡3y" 4- a"/3y' — d'(3'y, &c. 
(the law of whose formation is tolerably well known, but may be thus expressed, 
o j = a, 
a > ¡3 , 7 
<*', /3', 7' = a 
/3", 7" 
the signs 4- being used when the number of terms in the side of the square is odd, 
and 4- and — alternately when it is even.) 
Then the theorem in question is 
p a + a ¡3 + t y.., p o' + a /3' + r y.., p a" + a (3" + r y'.. 
p o + a (3 4- r y.., p d 
p"cl + a"(3 + r" y.., p"o! + a"13' + r"y.., p"a" + cr"/3" + r"y".. 
c.