A “ smith’s PRIZE ” PAPER ; SOLUTIONS. 
489' 
534] 
planes; the line in question is therefore a line of the quartic surface; and similarly 
the quartic surface contains each of the ten lines 123.456, 124.356,..., 156.234. We 
have thus in all 25 lines on the quartic surface. 
In the case of seven points 1, 2, 3, 4, 5, 6, 7, the locus is the curve of inter 
section of the quartic surfaces which correspond to the points 1, 2, 3, 4, 5, 6 and the 
points 1, 2, 3, 4, 5, 7 respectively: these have in common the ten lines 12, 13, 14, 15, 
23, 24, 25, 34, 35, 45 (which it is easy to see do not form part of the required locus), 
and they have therefore, as a residual intersection, a curve of the order 16 — 10, = 6, 
or sextic curve, which is the locus of the vertices of the cones which pass through 
the seven given points. 
14. Show that the envelope of a variable circle having its centre on a given conic 
and cutting at right angles a given circle is a bicircular quartic; which, when the given 
conic and circle have double contact, becomes a pair of circles; and, by means of the 
last-mentioned particidar case of the theorem, connect together the porisms arising out of 
the two problems: 
(1) given two conics, to find a polygon of n sides inscribed in the one and circum 
scribed about the other; 
(2) given two circles, to find a closed series of n circles each touching the two 
given circles and the two adjacent circles of the series. 
The equation of the given circle is taken to be 
(x - of + {y - /3) 2 = 7*, 
X^ 77“ 
and that of the conic 0 + tt = 1. This being so, we have a cos 8, b sin 8 as the 
a- b- ° 
coordinates of a point on the conic, which point may be taken to be the centre of 
the variable circle, and introducing the condition that the two circles cut at right 
angles, the equation of the variable circle is 
(x - a cos dy + (y — b cos 8)- = (a — a cos 8) 2 + (/3 — b sin 8) 2 — y 2 , 
that is, 
x? + y 2 — a 2 — /S 2 + 7 2 — 2ax cos 8 — 2by sin 8 = 0, 
where 8 is the variable parameter; and the equation of the envelope therefore is 
(x 2 + y- — cl- — /3 2 + y 2 ) 2 — 4a 2 it* 2 — 4 b 2 y 2 = 0, 
which is a quartic curve; and writing herein ^ | in place of x, y the equation would 
be of the second order in regard to x 2 + y 2 , z, and it thus appears that the curve has 
double points at each of the points x 2 4- y 2 = 0, z = 0, viz. that the envelope is a 
bicircular quartic. 
If the fixed circle touches the conic, then by a consideration of the figure it at 
once appears that the point of contact is a double point on the curve; and so if 
there is a double contact, then each of the points of contact is a double point on 
the curve. But in this case the curve is a bicircular quartic with four double points;, 
viz. it is a pair of circles. 
C. VIII. 
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