530 
ON POLYZOMAL CURVES. 
[414 
rj = 0 x f3z), &c., lie, it is clear, in the line ¡3% — ay = 0, which is one of the nodal axes 
of the curve. Similarly, in the second case, if 0 be determined by the foregoing 
equation, we may take as corresponding tangents through the two nodes respectively 
%=0az, y = — 0f3z\ the foci (A, B, G, D), which are the intersections of the pairs of 
lines (| =0 L az, 7] = — 0 1 /3z), &c., lie in the line ¡3^ + ay = 0, which is the other of the 
nodal axes of the curve. In either case the foci A, B, G, D lie in a line, that is, 
we have the curve symmetrical; and, as we have just seen, the focal axis, or axis of 
symmetry, is one or other of the nodal axes. 
149. In the case of the Cartesian, or when a = 0, ¡3 = 0, viz., the equation aa = b/3 
is satisfied identically, and this seems to show that the Cartesian is symmetrical; it 
is to be observed, however, that for a = 0, /9 = 0 the foregoing formulae fail, and it is 
proper to repeat the investigation for the special case in question. Writing a = 0, /9=0, 
the equation of the curve is 
+ ez 2 %r) + z 3 (at; + by) + CZ 4 = 0, 
and then, taking f = dbz for the equation of the tangent from 1, we have 
rj 2 . kb 2 0°- 
+ yz. b {e0 + 1) 
+ z 2 . ab0 + c = 0, 
and the condition of tangency is 
4k0 2 (ab0 + c) — (e0 + l) 2 = 0 ; 
viz., we have here a cubic equation. Similarly, if we have y = 0az for the equation 
of a tangent from J, then 
4tkcf> 2 (ab(f) + c) — (e$> + l) 2 = 0. 
Hence 0 being determined by the cubic equation as above, we may take 6 = 0, and 
consequently the equations of the corresponding tangents will be % = 0bz, rj = 0az, viz., 
the foci A, B, C will be given as the intersections of the pairs of lines {% = 0J)z, 
r] = 0 x az), &c. The foci lie therefore in the line a% - brj = 0; or the curve is symmetrical, 
the focal axis, or axis of symmetry, passing through the centre. 
Article Nos. 150 to 158. On the Property that the Points of Contact of the Tangents 
from a Pair of Goncyclic Foci lie in a Circle. 
150. We have seen that the sixteen foci form four concyclic sets (A, B, C, D), 
(A l5 B u G l} A), (^2> A, C 2 , A), (A 3 , B 3 , C 3 , A), that is, A, B, C, D are in a circle. 
We may, if we please, say that any one focus is concyclic—viz., it lies in a circle with 
three other foci; but any two foci taken at random are not concyclic 5 it is only a pair 
such as (A, Bi) taken out of a set of four concyclic foci which are concyclic, viz., 
there exist two other foci lying with them in a circle. The number of such pairs 
is, it is clear = 24. Let A, B be any two concyclic foci, I say that the points of 
contact of the tangents A I, AJ, BI, BJ, lie in a circle.