TABLE 3

DIVISION RATES IN SUCCESSIVE SIXTY-DAY PERIODS, ALL SERIES

to
10/15

11/2/'18

1/ 3

From this table it is evident that all series follow the same general history, and that, at corresponding periods of the life cycle, all have about the same vigor, although the actual dates may range through all twelve months of the year.

The experiments thus show not only that waning vitality leading to old age and natural death is manifested by Uroleptus mobilis, but also, that conjugation between two individuals at any stage of waning vitality, leads to a complete restoration of vitality.

FALKLANDIA

BY J. M. CLARKE

STATE MUSEUM, ALBANY, NEW YORK

Communicated, January 29, 1919

Falklandia is a name herewith applied to a continental land which, during the Devonian period in the occidental parts of the Southern Hemisphere, preceded Gondwana-Land and Antarctis. The history of Gondwana-Land is well established (Neumayr, Suess); its supposed earliest outlines have been approximately determined by the study of its land flora (D. White). The conception of Gondwana-Land is that of a great east-west southern continent which escaped the turmoil of world-wide postcarboniferous deformations and which continued its existence as a continental asylum for land and stream life till late in the Mesozoic time (Cretaceous) when incursions of the sea began which led to its breakdown and demolition in the Tertiary. Eastern Brazil into Sao Paulo, southern Argentine and the north half of the Falkland Islands constitute its western fragments; South Africa, the lost Lemuria (from Madagascar to Ceylon), India and Australia indicate its western extent. Those who have been responsible for the determination of this continent and especially Suess, who has discussed it in much detail, have not recorded its existence prior to the Carboniferous. Antarctis likewise, another Southern 'asylum,' defined on the basis of its terrestrial life and never accurately delimited by its proponents as to the date of its origin, gives proof of like beginning of stabilization and perhaps also of length of endurance. The fossil woods discovered in the Beacon sandstone of South Victoria-Land by James Eights ninety years ago, and the fossils brought home in recent years by Andersson, Nordenskiold, Amundsen, Shackelton and the men of Scott, tend to indicate that it was coexistent in time with Gondwana-Land.

Asylums, thought Suess, were to be defined by continuity in the succession of terrestrial life; it must be added, however, that security of such determinations can be given by the character of the life of the sea which washed the shores of such asylums. Gondwana-Land and Antarctis had a parallel existence in time, though a distinct one. Osborn's observations indicate the

breakdown of Antarctis in the Tertiary. Both Gondwana-Land and Antarctis 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 Devonian 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 received. 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 appropriately 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 Frech 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 Frech'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.

An abelian variety of genus p, V, belonging to an Sn, (n > p), is defined by equations

x; = f(U1, U2, · u„), (j = 1, 2, . . . n)

(1) where the (f)'s are 2p-ply periodic meromorphic functions of the (u)'s, the periods forming a matrix of a type called, after Scorza, a Riemann matrix. The rank of V, is the number of distinct systems of values of the (u)'s, modulo, corresponding to an arbitrary point of Vp. It is easy to show that to any corresponds a V, of rank one. Suppose indeed the (ƒ)'s so chosen that any other periodic meromorphic function belonging to be a rational function of them. If V, were of rank > 1 then to every set of values (1, น2, u,), would correspond another (uí, u2, . . . u), distinct modulo, such that (u) = (u') whatever. Choosing such functions with non zero jacobian we find that the (u')'s are functions of the (u)'s. But ❤ (u1⁄2 = (u1 — α1, up - αp) upap) whatever the constants since the function at the right is of same type as . Hence at once du1 = dun, un -up = B. The (8)'s are constants which obviously form a set of periods of the (f)'s, and therefore V, is effectively of rank one.-All abelian varieties of rank one belonging to are birationally equivalent.

α,

αι,

...

2. Denoting by m the complex conjugate of any number m, a real variety of Sm is defined by the condition that when it contains one of the points, said to be conjugate, (x), (x), it contains the other. Let then be a Riemann matrix with a corresponding real V, of rank one. Any simple integral of the

first kind of V, is of the form u = Sz R; dx;, where the (R)'s are rational in the

(x)'s. Replacing in them all coefficients by their conjugates we obtain a new integral of the first kind of V, say (u), and u is a linear combination of the two integrals u + (u),−i (u — (u)), which are of real form. Hence V possess p independent integrals of real form, u1, U2, . . . up. If V, has a real point and we take it for lower limit of integration, our integrals will assume conjugate values at conjugate points of V.

Let now y be any linear cycle of V, and 7 its transformed by T, transformation of the variety permuting each point with its conjugate. T transforms + into itself and y-7 into its opposite. As 2 y, = (x + y) + (x − y), the double of any cycle is the sum of two others transformed by T the one into itself and the other into its opposite. The periods of u with respect to cycles of the first type are real, and with respect to those of the other type they are pure complex. If q is the number of cycles of one type 2p - q is that of the cycles of the other type. As the real and complex parts of the periods of p independent integrals of the first kind with respect to 2p independent cycles form a non zero determinant, we must have = q= 2p-q, or

=

...

q= p. Finally we may single out 2p cycles Y1, 72, such that Y2p, T.Yh = Yh, T.Yp + h = — Yp + h, h≤ p while the cycles m11 + m2Y2 + . . . + m2pY2p, (m integer), include the double of any cycle of V. The corresponding period matrix for the (u)'s is of the type

Remark.—In all this V, could be replaced by any real irreducible algebraic variety of irregularity p.

...

...

3. Conversely if to a corresponds a V, of rank one with 2p linear cycles 71, 72, Y2p, such that the cycles m11 + M2Y2 + . . + M2pY2p include the double of any other, while p independent integrals of the first kind u1, U2,... up, have with respect to them a period matrix of type (2), V, is birationally equivalent to a real abelian variety. Indeed as a consequence of the assumptions made, if the equations (1) represent V, and if (w1, 2, ... wp) are a set of periods of the (f)'s so are (w1, 2,...wp). Moreover the (f)'s are real meromorphic functions of the (u)'s and of a finite number of constants. If we replace these constants by their conjugates we obtain new functions f(u1, U2, up) with the same periods as the (f)'s the equations

...

x'; = f; +ƒ¡, x}' = − i (f; — Î;), (j = 1, 2, . . . n)

represent in an S2n a real abelian variety birationally equivalent to V. This real abelian variety has ∞ real points.

4. Assuming then V, real with real folds, to determine their number we remark that at two conjuguate points the integrals u of No. 2 take conjuguate values uiu, uh- iu, modulo 2. At the real folds the (u)'s are given by uh+ Tuwhp+μ, (un arbitrary, r = 0 or 1; h = 1, 2, . . . p). There

1

are in general ps periods of this type such that no linear combination of them with integral coefficients is of the form

(mμ wh‚μ + im'μ wh‚p+μ), (h = 1, 2, . . . p).

Hence ps of the integers r can be made equal to zero. Taking the others equal to zero or one yields 2o distinct real folds.-That there are varieties having that number of real folds whatever sp is shown by the canonical matrix

Hence a real abelian variety can have any number of real folds given by

2o, 0 ≤s ≤ p.

5. Any real fold of the variety can be transformed into any other by a birational transformation belonging to the continuous group (un, un + c). Hence