PROFESSOR CAYLEY ON POLYZOMAL CURVES. 49 



points of (A, B) ), then A x , 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 anti-points 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-l), that isp 2 foci, the entire system 

 of foci. 



On the Foci of Conies — Art. Nos. Ill to 117. 



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 anti-points 

 A u B v 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 /, 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, A J, or say the 

 three sides of the triangle A IJ; 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 co-ordinates, and taking (a, a, 1) and (/3, (3\ 1) for the co-ordinates of the 

 given foci A, B respectively, the conic touches the lines £— az = 0, ri~az = 0, 

 £ — f3z = 0, t] — f$'z = ; the equation of a conic touching the first three lines is 



*/l(l — ccz) + Vm(% — /3s) -\- */n{n — a'z) = , 



where I, m, n are arbitrary, and it is easy to obtain, in order that the conic may 

 touch the fourth line n — (3'z = 0, the condition 



114. In fact, n having this value, the equation gives 



KZ-az) + m(£ - &) 4- 2A//m(g-«0(S-j&)= - f^A (m - *) (« - ?z + (0 - «>) , 



and taking over the term JpA (m- 1) (ft - a)z, = (/3 - «) (m - l)z, 



this gives 



l il ~ fr) + ™(l - «») + 2\Z/m(g-a*) (£-]&) = - |£^- (m - 1) (,- /3'z), 

 VOL. XXV. PART I. N 



