MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
573 
191. The coordinates (^£&>) of any circle which touches the curve at the point- 
{x'y'z'iv') must satisfy 
— 03 
_ ___ -f . 
(1 + 2 lS)x' —az' + 2ky' —ay'—lev— kz' 
The equation of the tangent at {x'y'z’iv’) will bo 
{2x'x-\-2 y'y—iv'z—wz){x cot r l — az cot i\—ay cosec e ) 
= {x'x-~az'y—ay'z){2x cot r l -\-2y cot r 2 —{id-\-z) cosec e) . 
The equation of the normal at {x'y'z'iv') will be, 
( 202 ) 
2x, 
2 y. 
IV, 
z 
= 0. . . 
. . (203) 
'a 
CM 
to 
ay 
w', 
z 
E, 
” az , 
ay', 
0 
2 cot i\, 
2 cot r 2 , 
cosec e, 
cosec c 
192. The system of bitangent circles is given by 
£=o, (y—a£f=& 
The focus of the curve is given by 
x y z 2 w 
0 a 2 a 2 
(204) 
(205) 
Equation of a Spheri-quadric Referred, to three Circles Orthogonal to one of its 
Principal Circles. —§§ 193-198. 
193. Let iv=0 be one of the principal circles of a spheri-quadric ; then if {x, y, z) 
are any three circles orthogonal to w, the equation of the spheri-quadric must be of 
the form 
w i -\-f{x, y,z) = 0, 
and the equation of the absolute will also be of the same form. Hence, by subtraction, 
we have for the equation of the spheri-quadric 
ax z -\-by‘ 2 -\-cz 2 -\-2fyz-{-2gzx-p2hxy=0, .(206) 
and this is a form to which the equation of any spheri-quadric can be reduced. 
194. We shall find it convenient to suppose the coordinates ( xyz) to be equal to the 
