[485 
485] PROBLEMS AND SOLUTIONS. 
569 
that the 
ly deter- 
hen this 
that is, 
■) 
i of the 
passing 
ee given 
;s drawn 
unicursal 
0, 0 = 0), 
equation 
which conic will touch the line through the points (a, ß, y) (a, ß', y), if 
*J\fW - &i)\ + \/{g (7 a ' - 7 a )} + (<*ß' ~ *'ß)} = 0. 
The equation of the pair of tangents from (a, ß, y) to the conic is 
(f\ g-, - gk - hf, - bg\yy - ßz, az - yx, ßx - ay)- = 0, 
« 2 (gy + hßf + y 2 {ha +fyf + 0 2 (fß + gay 
+ [Zgha 2 - (ha +fy ) (fß + ga )} 
+ 2*c {2hgß- - (fß + ga ) (gy + hß)} 
+ 2xy [Zfgy- - (gy + hß) (ha +fy )} = 0, 
but one of the tangents through (a, ß, y) being 
x (ßy — ß'y) + y (ya! — y a) + 0 (aß' — aß) — 0, 
it follows that the other tangent is 
r (gy + hßy 
ßy - ß'y 
+ y 
(ha +fyy 
ya! — y a 
1 (fß + g*y 
aß'-a'ß 
= 0. 
Hence, writing for shortness 
A =gy + h/3 , B =ha +fy, C =f/3 + ga, 
A' = gy + lt/3', B' = ha + fy, C' = f/3' + ga, 
the equations of the tangents from Q, Q' respectively are 
^ — + C 2 ' 
A 
A’ 2 
ßy - ß'y 
X 
-+B 2 
«ya — 7 a 
y 
- + B’ 2 —~ u — 7 + C' 2 
aß' — aß 
z 
T7>= 0, 
L5 = 0, 
/3y' — f3'y ya — y'a ' ~ a/3' — aft 
and for the coordinates of the intersection of these tangents, we have 
X y z 
ßy — ß'y ' y a! — y'a ' aß — aß 
: B 2 Ö'* - B'-C 2 : C 2 A' 2 - C' 2 A 2 : A-B'- - A' 2 B 2 . 
BC - B'G = f {-f(ßy - ßy) + g (ya! - y'a) + h (aß - aß)} 
BG' + B'G = 2ghaa' + f{ f(ßy + ß'y) + g (ya' + y a) + h (aß + a'ß)}. 
To satisfy the equation 
V {/(ßy ~ ßy)) + V [g (ya - y'a)} + ß [h (aß' - a!ß)}, 
write 
a 2 _ h 2 c 2 
J ßy - ß'y ’ y ya' — y'a ’ 1 aß' — aß ’ 
and therefore a 4- b + c = 0 ; we then have 
~f(ß7' “ ß'y) + g (y 01 ' ~ y' a ) + h ( a ß' ~ «ß)> = - « 2 + b- + c 2 , = - 26c ; 
C. VII. 
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