488 
a “smith’s prize” paper; solutions. 
[534 
The conclusion in regard to the areas of the polygons is that, taking k any given 
value whatever, however large, the ratio (m being of course > k) which the area between 
the ordinates to the abscissae m + k, m — k bears to the area of the whole polygon 
(or to unity) continually decreases as 2m increases, and ultimately vanishes; but con- 
trarywise, taking a any given fraction whatever, however small, the ratio which the area 
between the ordinates to the abscissae m + am, m — am bears to the area of the whole 
polygon (or to unity) continually increases as 2m increases, and ultimately becomes = 1. 
13. Show that for the quadric cones which pass through six given points the locus 
of the vertices is a quartic surface having upon it twenty-five right lines; and, thence or 
otherwise, that for the quadric cones passing through seven given points the locus of the 
vertices is a sextic curve. 
Suppose U = 0, V = 0, TF = 0, *8=0 are any particular four quadric surfaces passing 
through the six points, say 
(JJ= (a,...) (x, y, z, wf, V— (b, ...) (x, y, z, wf, &c.); 
then the equation of the general quadric surface through the six points will be 
all /3 T 7 " + 7 IF -fi 8S = 0, 
and this surface will be a cone, having (x, y, z, w) for the coordinates of its vertex, if 
only we have simultaneously 
dU i Q dV ^ dW s dS A 
+ + +S £ = 0 ' 
dU , 
a + &c. 
dy 
dU s 
a -f &c. 
dz 
dU p 
a —=— + &c. 
dw 
= 0, 
= 0, 
= 0. 
Eliminating (a, /3, 7, 8) we have an equation V = 0, where V is the Jacobian or 
d(U V W S) 
functional determinant ^ formed with the differential coefficients of the 
four functions ( U, V, W, S) : the locus of the vertex is thus a quartic surface. 
Calling the six points 1, 2, 3, 4, 5, 6, then taking as vertex any point in the line 
12, the lines from such point to the points 1 and 2 coincide with the line 12, and 
we can through this line and the lines to the remaining points 3, 4, 5, 6 describe a 
quadric cone; the quartic surface therefore passes through the line 12; and similarly 
it passes through each of the fifteen lines 12, 13, ..., 56. 
Again, taking 
and 456, we have 
the vertex anywhere in the line of intersection of the planes 123 
an improper quadric cone, viz. the plane-pair formed by these two