MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
537 
1111 
intersection; —, —, - the coordinates of the line at infinity; then the equation to 
the absolute will be 
x^-\-y z =izvj. 
The curve is clearly the inverse of a parabola. 
105. If {&){&>) be the coordinates of any circle touching the curve at the point 
(x'y'zw), we must have 
£ ij — 2co — 2£ 
(1 + k)x / —az! + ky' —ay'—2kw' —2 kz'' 
The equation to the tangent line at the point ( x'y'z w') will be, by § 72, 
1 , a , cl A, , , , , . 
—x - 2 — y )(xx -\-yy —2 zw —2 wz ) 
the equation to the normal being 
whence 
(77 —a£) 3 =4£w. 
107. The two double tangents are given by 
pf — 2 — 4:vp=0, 
—— — ——=0 ; 
r 2 e e 
the angle <£ between them being given by 
‘m-^y 
tan(/>= -^ % l/J 9 . 
r «.“ + 8 2 a 2 
(119) 
r, j r 
(l 1 z' + w’\ 
={7 x '+-y - 2 ——)( xx '- az 'y- az y') '> .( 12 °) 
X, 
y> 
-2 w, 
— 2 2 
= 0 . 
(121) 
X, 
y’> 
— 2 w, 
— 2z 
X, 
— az , 
-ay', 
0 
1 
1 
2 
2 
3 
0 
3 
^ 2 
e* 
c 
(£r)C M ) will be a 
bi tan gent circle, if k— - 
-1, in which case 
V 
K 
2(0 
1 
S' 
1 
2a> + ay 
~a 3 t 
as! -\-y r z' 
2 uf—ay' 
y 
2 w 
3 
( 122 ) 
(123) 
3 z 
MDCCCLXXXVI. 
