NOTES AND REFERENCES. 
597 
class of the envelope of the osculating planes of the cuspidal curve. Finally d denotes 
the number of stationary points (cusps) of the nodal curve, exclusive of the points 7 
which lie on the cuspidal curve; and g and e denote, g the number of ordinary 
actual double points of the cuspidal curve, e the number of stationary points (cusps) 
of the same curve, exclusive of the points /3 which lie on the nodal curve. 
Moreover with Zeuthen, the nodal curve has 
St +f+ 30' + 2' double points 
(k = k + St +/+ 30' + 27, if k denotes, as with me, the number of apparent double points 
of the curve), and it has 
7 + + 2' stationary points. 
The cuspidal curve has 
g + 6%' + 125' + XT + 40' + 2 + 2' double points 
(h = h + g + 6%' + 125' + U' + 40' + 2 + 2', if h denotes, as with me, the number of 
apparent double points of the curve), and it has 
/3 + e + 20' stationary points 
and the nodal and cuspidal curves intersect in 
3/3 + 27 + i + 120' +2 + 2' points ; 
where I have written 2 and 2' to denote sums (different in the different equations) 
determined by Zeuthen, and depending on the singularities (5 and S' respectively. 
For comparison of my formulae with Zeuthens it is thus proper in my formulae to 
write 0=0, a> = 0, 0 = 0 (but in the first instance I retain 6) and in his formulae to 
write U = 0, 0 = 0, d = 0, g = 0, e = 0, 2 = 0, 2' = 0. Doing this the last mentioned 
formulae give as with me 3t+f double points and 7 stationary points for the nodal 
curve, but they give for the cuspidal curve 6^' +12/3' (instead of 0) double points and 
/3 stationary points; and the two curves intersect (as with me) in S/3 + 27 + i points. 
There is a real discrepancy in the number 6^;'+ 12/3' of double points on the cuspidal 
curve.