ME. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
521 
Chapter VI. —General Equation of the Second Degree in Power- 
Coordinates. 
Nature of the Curve .—§§ 69, 70. 
69. The most general equation of the second degree in power-coordinates may be 
written 
( fi(x , y, z, w)=ax 2j rby' i -\-cz l -\-dw' 2 -\-2fyz-\-2gzx-\-2hxy-\-2lxw-\-2myw-\-2nziv=0, (93) 
(xyzw) being the coordinates of a point on the curve, referred to some system of 
circles, and therefore satisfying the equation of the absolute xfj, which is also of 
the second degree. Consequently the form (.93) contains only eight arbitrary 
constants. 
Now xyzw may be expressed as linear functions of X 3 +Y 3 , X, Y, 1; (X, Y) being 
the Cartesian coordinates of the point; and substituting, it is easily seen that (93) may 
be expressed in the form 
(X 3 +Y 3 ) 3 +U, (X 3 +Y 3 ) -b U 2 1= 0,.(94) 
U 1; XL being of the first and second degree; this equation contains eight constants, 
and since (94) represents a curve having nodes at each of the circular points at infinity, 
it appears that (93) is a form to which every bicircular quartic can be reduced. 
70. It is otherwise evident that, since the equation of a straight line is of the 
first degree, every straight line cuts (/> in four points, unless <j> is satisfied by the 
coordinates of the line at infinity ; for these coordinates satisfy the equation of every 
straight line, and therefore in this case < f> must represent a circular cubic mid the line 
at infinity. 
Equation to Tangent at any Point .—§§ 71, 72. 
71. Let be the coordinates of any circle touching the curve at the point 
[xyzw). We must have, by equation (74) 
d -± w i v > i d ± z > . w '- a 
+ d v J + 8o> ~ U ’ 
and since this passes through the point [x + 8x, y + §y, z -\-8z, w -f- 8 w ) w 7 e must 
have 
cty , 8V *„./ , 3^ ?>,,/ i ^ , 
COD 
3 x 
' dG , |°f v '—A 
Ac. S* ~hw oy -hmr §2 -f- ^ ow — 0 , 
crj OQ rr ' 
MDCCCLXXXVI. 
