537] 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
505 
■»'«* • rr... 
Taking the conics to pass through the point x= 0, y = 0; their equations will be 
of the form 
X = a^x? + 2h±xy + b x y 2 + 2\fiyz + 2 g x zx = 0, 
Y = a 2 x 2 + 2 h 3 xy + b 3 y 2 4- 2f 3 yz + 2g 3 zx = 0, 
= a 3 x 2 + 2h 3 xy + & 3 y 2 + 2f 3 yz + 2g 3 zx = 0, 
W = a 4 x 2 + 2h 4 xy + b A y 2 + 2f 4 yz -f 2g 4 zx = 0. 
Now multiplying by the indeterminate quantities a, ¡3, y, 8, the three ratios a : /3 : y : 8 
may be determined so that the terms in yz, zx shall vanish, and the terms in a?, xy, y 2 
be a perfect square: we thus arrive at a quadric equation for any one of the ratios, 
say a : ¡3, the remaining ratios being then linearly determined; viz. there are two sets 
of values of a, (3, y, 8: and changing the coordinates (x, y), the two resulting forms 
may be represented by x 2 =0, y 2 — 0. 
And it is clear that we thus have in the system of conics aX + (3Y+ yZ + 3 W = 0, 
four conics the equations of which may be represented by 
= 0, 
= 0, 
Z' = h 3 xy +f 3 yz +g 3 zx = 0, 
w = Jhxy +f 4 yz + g 4 zx = 0, 
where of course the coefficients f, g, h have new values. 
We may then form the equations 
aX' +f,Z' —f 3 W' =x {ax +(f 4 h 3 -fA) y + (f 4 g 3 -f 3 g A ) zj, 
/3Y' - g 4 Z' + g 3 W'=y {(g 3 h 4 - g 4 h 3 ) x +/3y + (f 4 g 3 -f 3 g 4 ) zj, 
so that, by writing a = g 3 h 4 — g 4 h 3 and ¡3 —f 4 h 3 —f 3 h 4 , the terms in { } will be one and 
the same linear function of (x, y, z); that is, changing the z so as to denote the 
linear function in question by £, we have as conics of the series xz = 0, and yz = 0, 
that is, we have in the series the four conics x 2 = 0, y 2 = 0, xz = 0, yz= 0; whence also 
any other conic of the series, and consequently each of the original four conics, may 
be represented by an equation of the form 
ax 2 + by 2 + 2fyz + 2gzx = 0. 
7. The coordinates (x, y, z, w) of a point P in space are connected with the 
coordinates (x, y', z) of a point P' in a plane by the equations 
x : y : z : w = X' : Y' : Z' : W', 
where X', Y', Z', W are quadric functions of (x', y', z r ) such that X' = 0, Y' — 0, Z' = 0, 
W' = 0 represent conics having a common point: show that the locus of P is a cubic 
scroll {skew surface of the third order): and find the curves in the plane which corre 
spond to the generating lines of the scroll. 
C. VIII. 
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