PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
667 
A, B, C . 
. K 
B, C, 1) . 
. L 
( 1 ) 
where the A, B, C . . . K, L are linear functions of the coordinates, and the number 
of columns is —n. For the n -\- 1 homogeneous coordinates are proportional to 
rational integral functions of t of the n th order. Solving these n+ 1 equations for 
1, t, ft . . . t n we find 
1 , t, t* . . . t n = A, B, C . . . L, 
which is equivalent to the system written down above. 
The more general system of equations — 
A, B . 
. K 
A', B' . 
. K' 
( 2 ) 
where the A ... K, A' ... K' are linear functions as before, may always and easily be 
reduced to the former, for they are got by eliminating t from the n equations. 
A+^A — 0, (3) 
B + *B' = 0, 
K + *K'=0. 
We may, however, solve these equations for the ratios of the coordinates, which 
will thus be expressed as rational functions of t of the n a order. Solving these for 
1, t, t 2 . . . t n we come back to the previous system. 
The equations (3) exhibit the curve as the locus of the intersection of corresponding 
elements in n projective pencils. 
(n— l)flat 
which 
passes 
thr 
easily seen to be 
0 = 
A, B, 
C, ... 
L 
1) A 
t 3 
H l 
i ? 
t 2 
L n 5 • • • 
t n n 
But this equation is manifestly divisible by the coefficient of L, which is the product 
of the differences of all the t. If we write — 
2i = *i+*a+*3+ 1 ' ' AA? 
'£%=hh+ t \h+hh+ • • • 
etc. = etc. 
S — t t / 
then the equation is 
0=L-K^+. . .diBS^AS, 
(4) 
