48 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



we have q = 2 ; if it be a node, each of the tangents count twice, or q — 4 ; if it 

 be a cusp, the tangent counts three times, or q = 3. Similarly, if the other 

 circular point at infinity be an ordinary or a singular point on the curve, the 

 tangent or tangents there count a certain number of times, say q times, among the 

 tangents to the curve from this point ; the number of the remaining tangents is 

 thus = n —r/. And if as usual we disregard the tangents at the two points /, J 

 respectively, and attend only to the remaining tangents, the number of the foci 

 is = (n — q) (n — q') . 



107. Among the tangents from the point I or J there may be a tangent which, 

 either from its being a multiple tangent (that is, a tangent having ordinary con- 

 tact at two or more distinct points), or from being an osculating tangent at one 

 or more points, counts a certain number of times, say ;•, among the tangents from 

 the point in question. Similarly, if among the tangents from the other point 

 J or I, there is a tangent which counts / times, then the foci are made up as 

 follows, viz. we have — 



Intersections of the two singular tangents counting as 

 Intersections of the first singular tangent with each 



of the ordinary tangents from the other circular 



point at infinity, as .... 

 Do. for second singular tangent, 

 Intersections of the ordinaiy tangents, . . (n — q — r){u — q'—r') 







r'r 



foci 



(«- 



-'/- 



-r')r 



»» 



(»- 



-q- 



-;•)/ 



• • 



Giving together the {n—q) (n — q) foci : 



and the like observation applies to the more general case where the tangents from 

 each of the points 7, •/ include more than one singular tangent. 



108. There is yet another case to be considered ; the line infinity may be an 

 ordinary or a singular tangent to the curve : assuming that it counts s times 

 among the tangents from either of the circular points at infinity, the numbers 

 of the remaining tangents are n — q — s, n — q'— s from the two points /, J 

 respectively, and the number of foci is = (n — q — s)(n — cf— s) . 



109. In the case of a real curve the two points I, -/are related in the same 

 manner to the curve, and we have therefore q = q ; the singular tangents (if any) 

 from the two points respectively being the same as well in character as in num- 

 ber. Writing n — q — s — n — </ — s, = ^>, and not for the present attending to 

 the case of singular tangents, I shall assume that the number of tangents to the 

 curve from each of the two points is = p ; the number of foci is thus = \f ; and 

 to each focus there corresponds a directrix, viz., this is the line through the points 

 of contact of the tangents from the focus to the two points /, J respectively. 



110. Consider any two foci A, B not in lined with either of the points 7, J, 

 then joining these with the points /, J, and taking A^ B x the intersections of 

 A I, BJ and of A J, BI (A l9 B r being therefore by a foregoing definition the anti- 



