MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
529 
Chapter VII.— Classification of Bicircular Quartics. 
i. Method .—§ 85. 
85. It will be convenient to take as the basis of our classification the nature of the 
roots of the discriminating quartic; we shall thus have three species, each of which 
may be subdivided into two—according as the double points at infinity are nodes or 
cusps. We shall then have three similar species of circular cubics. 
Using the notation employed by Salmon (‘ Higher Plane Curves,’ § 82), we shall 
denote the characteristics of a curve by (to, n, 8, r, k, i), and we see that we shall have 
the following cases :— 
m 
n 
£ 
T 
1C 
i 
Name. 
i. 
4 
8 
2 
8 
0 
12 
ii. 
4 
6 
0 
1 
2 
8 
Cartesian 
iii. 
4 
6 
3 
4 
0 
6 
iv. 
4 
4 
i 
i 
1 
2 
2 
Li macon 
V. 
4 
5 
2 
2 
1 
4 
vi. 
4 
3 
0 
0 
3 
0 
Cardioid 
vii. 
3 
6 
0 
0 
0 
9 
viii. 
3 
4 
1 
0 
0 
3 
ix. 
3 
3 
0 
0 
1 
1 
(i.) may be called the general bicircular quartic; (iii.) is the general inverse of a 
conic; (v.) is the inverse of a parabola; (vii.) may be called the general circular 
cubic; (viii.) is the inverse of a conic with respect to a point on the curve; and (ix.) 
is the inverse of a parabola with respect to a point on the curve. 
General Bicircular Quartic. — §§ 86-92. 
86. The equation of the curve may be written 
ax 2 + b/f- +ez 2 + did 1 = 0; 
and if we write the absolute 
xJj= x 2 + y 3 •+ z 3 + iv 2 = 0, 
the coordinates of the line at infinity will be —, —, —, i.e., the reciprocals of the 
... . . r i r 2 r 3 T i 
radii of the principal circles. 
87. The coordinates of any tangent circle at the point (x'y'ziv) will be proportional to 
{ci-\-TQx , (b-\-k)y , (c-\-TQz , {d-\-JQw. 
3 Y 
MDCCCLXXXVI. 
