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SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
[537 
The equations x : y : z : w = X' : Y' : Z' : TT' are three equations containing the 
indeterminate parameters x : z' and y' : z, so that eliminating these we have between 
(x, y, z, w) a single (homogeneous) equation representing a surface. To each point 
{x\ y’, z') of the plane, there corresponds a single point of the surface, and to each 
point (x, y, z, w) of the surface a single point {pc', y, z) of the plane. The only- 
exception is that for the common point of the four conics, the ratios x : y : z : w 
are essentially indeterminate, and there is not corresponding hereto any determinate 
point of the surface. 
To find the order of the surface, consider its intersection with any arbitrary line 
ax + by + cz + div = 0, 
a x x + b x y + C\Z + d x w = 0. 
We have corresponding hereto in the plane the points of intersection of the conics 
aX' + ÒY' +cZ' +dW' =0, 
a 1 X' + b 1 Y' + c 1 Z' + d 1 W' = 0, 
viz. these are conics each of them passing through the common point of the four 
conics, and therefore they intersect besides in three points : that is, the order of the 
surface is = 3. 
To show that the common point ought to be (as above) excluded, some further expla 
nation is desirable. To the section of the surface by the plane ax + by + cz + dw = 0, 
corresponds the conic aX’ + bY' + cZ' + dW = 0 ; and similarly to the section by the plane 
a 1 x + b x y + c x z + d x w = 0, corresponds the conic a x X' + b x Y' + c x Z' + d x W = 0. Now to the 
common point considered as belonging to the first conic there corresponds a determinate 
point of the surface ; and to the common point considered as belonging to the second 
conic there corresponds a determinate point of the surface ; but these are two distinct 
points on the surface : so that corresponding to the common point of the four conics, 
there is not on the surface any point of intersection of the two plane sections ; but 
these intersect in only three points of the surface ; viz. the line of intersection of 
the two planes meets the surface in three points : or the surface is a cubic surface. 
The same result may be obtained, and it may be further shown that the surface 
is a scroll, by means of the property in the foregoing question 6 ; viz. it thereby 
appears that each of the functions X', Y', Z', W may be taken to be of the form 
ax* + by' 2 +fy'z'+ gz'x ; hence replacing the original coordinates x, y, z, w, by properly 
selected linear functions of these coordinates, the given relations may be presented in 
the form 
x : y : z : w = x' 2 : y' 2 : x'z : y'z, 
whence eliminating, we have 
xw 2 — yz 2 = 0 
the equation of a cubic scroll, having the line z = 0, w = 0 for a double line, and the 
line x = 0, y = 0 for a directrix line. The equations of a generating line of the scroll