258 
of the first kind ($ 6) cut e‚ in Z,; the number of intersections, 
arising from them, amounts therefore to : 
2 (u—2e — 20) (u—2e—20-—1). 
The 40 (o—-1) branches of the second kind, and the 40 (u-—2e—2o0) 
of the third touch ¢, on the contrary in Z,. and so give respecti- 
vely 86(6—1) and 86 (u—2e—2o)-intersections ; the sum total of these 
three numbers is: 2g?-—Sue—2u+8¢e?+4e—4o. 
If this number is subtracted from the order of the rest nodal, 
curve as given in § 6, the number of points ¢, has in common 
with the rest nodal curve apart from 7, is found; this numher 
amounts to 
o = 4uv—4uo — 8ve+ 826— 13+ 8e + 90 + 54 v* + 6* —2v6+-3t. 
These points lie in pairs harmonically with regard to Z, and 9, 
either on #2, or on the generatrices of 2 lying in e‚ and passing 
through Z,: their projections on /, are points at infinity of the locus 
of the points of equal tangents at 4”. To these points at infinity 
however belong also the projections of those points lying infinitely 
near to Z,, and which we have already determined in the prece- 
ding §, viz. 26(o—1)} simple, and (y--;2e—2o) 2c-fold ones (the latter 
lying in the simple intersections of k” and /,). If these numbers are 
doubled, are then added to the given number g mentioned above, 
and the result is divided by 2, the order d* of § 6 is exactly found back. 
If a branch of the rest nodal curve of 2 gets into /,,, and if we 
let a point P describe that branch, and that in the direction towards 
J?, the image circle of P, which twice cuts 4” perpendicularly gets 
greater and greater: if P moves along a hyperbolic branch, 
that circle will have as limit a straight line containing the inter- 
section of the asymptote with g, and this straight line will twice 
cut ke perpendicularly, and consequently be a double normal. If, 
however, P moves along a parabolic branen, the circle will in 
the end disappear into infinity. In this way the number of double 
normals of % is therefore not to be determined; this, however, is 
not necessary, as we already determined this number pretty nearly 
in our former paper. (Anw. Cykl. p. 21). The double normals of 
hk” are namely apparently double tangents of the evolute of 4“, and for 
the number of these double tangents we found ibid. : 
Wu 4+ »—2e— 0) (u4+ p—2e—o—l) — («+ Su)j. 
In this number /,, however, is comprised a number of times. 
The evolute namely has cusps in each of the «¢—2s«—2e simple 
intersections of 47 with /,, and that in such a way that the 
D 
cuspidal tangent coincides with /,, and in each of the 5 points of 
