ON POLYZOMAL CURVES. 
515 
414] 
I will however give the investigation in this simplified form, for the identity 
— ¿21 + + n(E = — ¿x2l + ; viz., in this case we have 
{ = (ft ~ 7) (ft -J>) _ n (ft - 7) (7 -&) 
a (3 (y — a) (a — B) y (a - /3) (a - 7) ’ 
and the identity to be satisfied is 
-l (£-«* )(»?--**) = -¿1 (f-«*)(*?- 1«) 
+ m (^ - fts) ~ (I- ft*) («7 “ 
+ » (£ - 7s) ^77 - 1 ^ +Wi (£ - 7*) («7 - ; 
writing %=az, 7) = -pZ, we find m 1} and writing £ = o.z, 77 = — ^r, 
then easy to obtain the value of l 1} viz., the results are 
we find n 1} and it is 
h _ m (a - /3) (ft - 7) n (ft — 7) (7 — tt ) m . __ 7~ g _ _ _ m g ~ ft 
S /3 (7 — a) (a— S) y (a — /3) (a— S) ’ 1 a—/3 ’ 1 7—a 
and therefore nn x n^ — mn\ it may be added that we have 
4 _ ft - 7 /mi wA 
r«-H7 + S/’ 
viz., this is the form assumed by the equation 
I m, ?i, 
- + TT+- 
a x b x Cj 
= 0. 
Part III. (Nos. 105 to 157). On the Theory of Foci. 
Article Nos. 105 to 110. Explanation of the General TJteory. 
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 Pliicker, 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 /, 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 n- foci. It is to be added that, as in the conic the line joining the points 
of contact of the two tangents from a focus is the directrix corresponding 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. 
65—2