( 832 ) 



are formed, on account of (he early degeneration of the nucleus, 

 which, by its divisions should have given rise to these nuclei. Further 

 more, after the egg-apparatus has been formed, the remaining portion 

 of the embryo-sac is only very slightly developed, so that there is no 

 question of the formation of endosperm (what happens to the second 

 generative nucleus, if indeed present, I have not been able to make 

 out). It is much clearer here than in most cases, that this portion 

 of the embryo-sac and the egg-cell are sister-cells. This agrees with 

 the view of Porsch ^), according to whom the egg-apparatus of the 

 higher plants is a reduced archegonium, the synergids being the neck 

 canal-cells and the upper part of the embryo-sac with the upper 

 polar nucleus being the ventral canal-cell. The latter hypothesis is 

 however specially difficult in this case, for here the positions of egg- 

 cell and of ventral canal-cell would be exactly reversed. A reduction 

 in the antipodal apparatus, similar to that which occurs here, is found 

 in Helosis guyanensis, according to the investigations of Chodat and 

 Bernard"), and a still further reduction exists in Q//9ri)9^(im?/z, where, 

 according to the researches of Miss Pace'), the lower portion of the 

 embryo-sac has not even been laid down at all. It need scarcely be 

 argued, that we are here concerned with a progressive differentiation, 

 and not with the recurrence of ancestral peculiarities. Perhaps it 

 may not be amiss to point out, in conclusion that we cannot here 

 fall back for "explanation" on a parasitic or saprophytic mode of life 

 of Podostemaceae. 



Mathematics. — ''On twisted curves of genus two". By Prof. 

 J. DE Vries. 



1. A curve of genus tiro bears one and only one involution of 

 pairs of points P. On the nodal biquadratic plane curve it is deter- 

 mined by a pencil of rays, having the node as vertex; its coincidences 

 are then the points of contact of the six rays touching the curve. 

 If we could arrange the points of the curve in a second 7' then 

 this ƒ' would be projected out of the node by a system of rays with 

 correspondence [2], and the above six tangents would furnish six 

 rays of ramification whilst a [2] can have four only. 



1) 0. Porsch. Versuch einer phylogenetischen Erklarung des Embryosackes und 

 der doppelten Befruchtung der Angiospermen. Jena 1907. 



2) R. Chodat el G. Bernard. Sur le sac embryonnaire de I'Helosis . guyanensis. 

 Journal de Botanique. T. XIV. 1900. p, 72. 



3) LuLA Pace. Ferlilizalion in Gypripedium. Botanical Gazette. XLIV. 1907. p. 353. 



