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7. If, still imagining that the cone-vertices S and 7’ are united in 
a point of the surface of {H], we suppose the cone [S| to be such 
that it touches the tangential plane in S to [|M], then 7* gets a cusp 
in S and R coincides with S and 7’, so that through #* pass but 
two quadratic cones; the plane RS 7’ remains determined, as the 
polar plane of H with respect to the cone [Rk] = [S]= [7], 
and in this plane lies the tangent in the cusp 2. The central projection 
now becomes a £* with three cusps, and the cuspidal tangent in 
R' is the line A. The double tangent through H remains determined 
as a trace of the plane of projection with the second tangential plane 
through O to [MH]; however, as is easy to prove from the stereometric 
diagram, the points of contact must necessarily be imaginary. 
As k* is situated harmonically with respect to h and H, the tangents 
in the cusps O,, O, must intersect each other on 4; however, / is 
the tangent in the cusp &': so the three cuspidal tangents pass 
through one point. 
The point H forms with the cusps O,,9,, and the point of 
intersection of the line O,O, with 4 a harmonic group; but now 
the same must hold for the two other sides of the AO,O,R’ of the 
cusps; the double tangent d is therefore the so-called harmonical line 
of the points uf intersection of the three cuspidal tangents with respect 
to the triangle of the three cusps; and the points of contact of the 
double tangent are the nodes of the elliptic involution on d, of which 
the points of intersection of d with the sides of that triangle and 
with the cuspidal tangents in the opposite vertices are three pairs. 
With respect to each cuspidal tangent and the point of intersection 
of the opposite side of the triangle with it the curve lies harmonically 
with itself. 
Zoology. — “On Ptilocodium repens a new Gymnoblastic Hydroid 
epizoic on a Pennatulid.” By Miss Winierep E. Cowarp B.Sc. 
Victoria University of Manchester. (Communicated by Prof, 
Max WEBER). 
(Communicated in the meeting of January 30, 1909.) 
Order. Gymnoblastea — Anthomedusae. 
Family. Ptilocodiidae. fam. nov. 
Ptilocodium repens: gen. nov., sp. nov. 
“Siboga” Expedition Stat. 289. 9° 0,3 5. 126° 24,5 E. 112 metres. 
Among the Pennatulids sent to Professor Hickson from the Siboga 
