MATHEMATICS: S. LEFSCHETZ 
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breakdown of Antarctis in the Tertiary. Both Gondwana-Land and Ant- 
arctis had a far longer duration than any of the continents of today. 
In the period immediately preceding the isolation of these continental 
masses they were united at the west; that is, in the occidental South Atlantic, 
the south polar land extended continuously into the land regions of the 
Gondwana Continent. This we know from the determinations of the Devo- 
nian strand lines in southern South America, the Falkland Islands and 
South Africa. 
The Devonian of these latitudes is a unit both in life and in sedimentation. 
In this regard it is wholly unlike the Devonian of Eria, the east-west continent 
of the North, and it is a conclusion that is irrefragable on the basis of the 
intimate and refined analysis that such determinations require and have re- 
ceived. The haphazard observer may be blind to these radical distinctions, 
especially when basing interpretations upon a knowledge of the strand faunas 
of Eria. The known extent of these Southern Devonian shore faunas, as 
pointed out by the writer (Fosseis Devonianos do Parana; Monographias, 
Vol. I, Servico Geol. e Mineral do Brazil, 1913), indicates the union of the 
Gondwana and Antarctis continents throughout the Devonian. The extent 
of this Devonian land bridge across the Atlantic is clearly shown by the 
unity of shore faunas in South Africa, Sao Paulo, Argentine and Bolivia, and 
the indication is of a land composed of Paleozoic strata of still earlier date. 
This is Falklandia, the parent land asylum out of which, in Postcarboniferous 
time, western Gondwana and Antarctis were carved. The name is appro- 
priately taken, for on the Falkland Islands the Devonian marine strata 
border the Gangamopteris (Glossopteris) beds of Gondwana-Land. 
Other names which have been suggested for these pre-Gondwana austral 
lands have been founded on inadequate evidence. The "South Atlantic 
Island" of Freeh indicated a Devonian land which had no connection with 
South Africa; Katzer imagined a north-south Devonian Atlantis running 
along the axis of the ocean, and Schwarz drew, with somewhat freer hand, his 
" Flabellites Land," as an undivided land mass along the Atlantic axis reaching 
from the north, and at the south spreading west and east to join Freeh's 
''South Atlantic Island." 
ON THE REAL FOLDS OF ABELIAN VARIETIES 
By S. Lefschetz 
Department of Mathematics, University of Kansas 
Communicated by E. H. Moore, January 30, 1919 
1. In this note we propose to find the number of real folds that an abelian 
variety of rank one may have and to establish some simple properties regarding 
their connectivity. 
