PROFESSOR CAYLEY ON POLYZOMAL CURVES. 47 



and the identity to be satisfied is 



- Z(f - «) (% - ^ = - h(l - m)(% - jja) 



+ »(S - 7-') (" ~ ~ z ) + "i^ - 7 s (« ~ jg*) ; 



writing ^ = a^, ^= -2, we find 7W X , and writing £ = «£, n = - 2;, we find n Y , and 



p / 



it is then easy to obtain the value of k , viz., the results are 



h = m (a - /3) (/3 - 7 ) w (0 - y)(y - «) _ _ ,„y - <* _ _ « - /3 



5 |8 (y - a) (a - b) y (o - j8) (a - 3) ' * a - |8 ' 1 y - a 



and therefore m^ — mn\ it may be added that we have 



5 a — 5 \ y 3/ 



viz., this is the form assumed by the equation -^ + -y- 1 + -i = .. 



<&•% D| C-j 



Part III. (Nos. 105 to 157.) — On the Theory of Foci. 

 Explanation of the General Theory — Art. Nos. 105 to 110. 



105. If from a focus of a conic we draw two tangents to the curve, these pass 

 respectively through the two circular points at infinity, and we have thence the 

 generalised definition of a focus as established by Plucker, viz., in any curve a 

 focus is a point such that the lines joining it with the two circular points at 

 infinity are respectively tangents to the curve; or, what is the same thing, if 

 from each of the circular points at infinity, say from the points I, J, tangents are 

 drawn to the curve, the intersections of each tangent from the one point with each 

 tangent from the other point are the foci of the curve. A curve of the class n 

 has thus in general ri l foci. It is to be added that, as in the conic the line join- 

 ing the points of contact of the two tangents from a focus is the directrix cor- 

 responding to that focus, so in general the line joining the points of contact of 

 the tangents from the focus through the points I, J respectively is the directrix 

 corresponding to the focus in question. 



106. A circular point at infinity / or J, may be an ordinary or a singular 

 point on the curve, and the tangent at this point then counts, or, in the case of 

 a multiple point, the tangents at this point count a certain number of times, say 

 q times, among the tangents which can be drawn to the curve from the point ; 

 the number of the remaining tangents is thus = n — q. In particular, if the 

 circular point at infinity be an ordinary point, then the tangent counts twice, or 



