672 
PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
0 = 
0 , (12)'*, (13)", . . . (1 n) n 
(21 )*, 0 , (23)", . . . (2 n) n 
(31)*, (32)", 0 , . . . (3 n) n 
(nl) n , (n2) n , ( nS ) a , ... 0 
This is obviously always satisfied if n is odd, for then the determinant is skew 
symmetrical, and being of odd order it necessarily vanishes. If, however, n is even, 
the determinant is a symmetrical function of the roots which vanishes when any two 
of them are equal ; and consequently it must contain as a factor the product of the 
squares of their differences. Now the determinant is of the order 2 n in each root, 
and the discriminant is of order 2 (n — 1) ; therefore the remaining factor is of order 2 
in each root, and being a symmetrical invariant must be a function of the squares of 
them differences. It can therefore be no other than X(cq — a 3 ) 3 (a 3 — a,) 3 . . . (a„_ x — a ,) 2 ; 
this is, to a factor pres, equal to the quadrin variant of the form (t — cq)(£ — a . 2 )■••(*-«.). 
The equation to the (n — 2)flat passing through two consecutive points of the curve 
at t, and through n — 3 other points p q ... u, is clearly 
0 = | x dx p q ... y 
I 1 2 3 4 ... n 
where the y are current coordinates, and the determinant is expressed in umbral 
notation. Writing in this for ay, (t — a,)'*, and for dx;, n(t — a ,)"~ l dt, we may observe 
that the determinant 
(t — cq) ra , (t—a 2 ) n 
(t — cto )" -1 
(cto — cq)(£ — cq)" l (t — a 2 )" T , 
so that the equation is of order 2 (n — 1) in t. It thence follows that 2 (n — 1) different 
(n — 2)flats may be drawn through n— 2 arbitrary points to touch the curve; or that 
the developable traced out by the tangent lines is of the order 2(w — 1). 
Similarly, from the value of the determinant 
{t~c q)" , {t-a 2 ) n , . . . {t-a k+1 y 
(t- ai y-\{t- oo)*- 1 ,. . . {t-a k+ 1 y~ i 
. . (t-a Hi y~* 
whicli is equal to the product of the differences of cq, a 3 , . . . a k+l multiplied by 
{ (t — o-i) {t — a 2 ) . . . {t-a Hl )} n ~\ 
we may conclude that the number of (n — 2)flats which can be drawn through k con- 
