OX POLYZOMAL CURVES. 
517 
414] 
110. Consider any two foci A, B not in lined with either of the points I, J, 
then joining these with the points I, J, and taking A 1} B x the intersections of AI, BJ 
and of AJ, BI (JLj, B 1 being therefore by a foregoing definition the antipoints of (A, B)), 
then A 1} B x are, it is clear, foci of the curve. We may out of the p 2 foci select, and 
that in 1.2. .p different ways, a system of p foci such that no two of them lie 
in lined with either of the points I, J; and this being so, taking the antipoints of 
each of the %p(p — 1) pairs out of the p foci, we have, inclusively of the p foci, in 
all p + 2. \p (p — 1), that is p 2 foci, the entire system of foci. 
Article Nos. Ill to 117. On the Foci of Conics. 
111. A conic is a curve of the class 2, and the number of foci is thus = 4. 
Taking as foci any two points A, B, the remaining two foci will be the antipoints 
A 1} B 1 . In order that a given point A may be a focus, the conic must touch the 
lines AI, AJ; similarly, in order that a given point B may be a focus, the conic 
must touch the lines BI, BJ; the equation of a conic having the given points A, B 
for foci contains therefore a single arbitrary parameter. 
112. In the case, however, of the parabola the curve touches the line infinity; 
there is consequently from each of the points I, J only a single tangent to the 
curve, and consequently only one focus: the parabola having a given point A for its 
focus is a conic touching the line infinity and the lines AI, AJ, or say the three 
sides of the triangle AIJ; its equation contains therefore two arbitrary parameters. 
113. Returning to the general conic, there are certain trizomal forms of the focal 
equation, not of any great interest, but which may be mentioned. Using circular 
coordinates, and taking (a, a', 1) and (/3, ¡3', 1) for the coordinates of the given foci 
A, B respectively, the conic touches the lines £ — az = 0, p — a'z — 0, ^ — ¡3z = 0, 
p _ fi'z = 0; the equation of a conic touching the first three lines is 
fl (£ — az) + Vm (£ — (3z) + fn (p — a'z) = 0, 
where l, m, n are arbitrary, and it is easy to obtain, in order that the conic may 
touch the fourth line p — /3'z = 0, the condition 
/3 — a 
n = — 
P-a‘ 
> (m ~ 0- 
114. In fact, n having this value, the equation gives 
l (£ - az) + m (£ - /3f) + 2 'Jim (g - az) {£ - /3z) = - ( m — 0 0? “ $' z + (£' - a') z), 
and taking over the term 
(fi'-a^z, =(/3 -a)(m-l)z, 
this gives 
l (f — /3z) + m (| - az) + 2 flm (f - az) (£ - ¡3z) = - _ (m — l){p— ¡3'z),