Lapas attēli
PDF
ePub

nomena are to be regarded as saturated solutions of a rigid phase, carbon, in a fluid phase, gas. It suggests that repeated solution and subsequent evaporation, here taking the form of adsorption at low temperatures and outgassing at high, produces a gradual modification of the character of the carbon with respect to its fineness of division, something similar to precipitation occurring. Loss of activity seems difficult to account for on the first hypothesis. It is to be noted that permanent loss of activity can always be produced on any sample by heating to 1200°C. This is usually ascribed to a partial destruction of porosity and has been observed before. Attempts to extract heavy hydrocarbons by the use of the lightest liquid solvents, ligroin alcohol and acetone were inconclusive. Considerable amounts of tarry material were removed, but not by the solvents directly. They distilled out during outgassing after treatment with the solvent. This treatment in all cases causes temporary loss of activity which is renewed in the usual manner with repeated use after low temperature outgassings. Experiments by others however' which have shown that activation is possible by other methods than the ones here outlined seems to favor the hydrocarbon hypothesis.

Those other methods for activation of field material for the adsorption of complex vapors have been used in conjunction with the above described process. A comparison of material activated by use and low temperature outgassings alone as herein described, with the most highly activated charcoals produced elsewhere under the conditions of these experiments is given in figure 6. The crossing of the curves may be very significant. The laboratory charcoal, R. 16, is less active initially but ultimately runs to lower values than the field material U. S. 4 and U. S. 6. The differences at either end are not large in comparison with the range of the phenomena discussed above. Experiments are in progress on saturation values for adsorptions of mixtures of varied proportions. An hypothesis originally advanced by McBain that there is a distinction to be made between surface condensation and interior diffusion is also being subjected to experimental scrutiny with modern materials now at our disposal. Both of these lines of work it is hoped will shed light on the mechanism of the process.

A more detailed account of this work will shortly appear in the Physical Review as a series of papers under the general title of Studies in Charcoal Adsorption.

1 This article is published with the approval of Major General William L. Siebert, Director Chemical Warfare Service, U. S. A.

Shrader, Ithaca, N. Y. Physic. Rev., 12, 1918, (70).

3 Homfray, Liepzig, Zs. Phys. Chem., 74, 1910, (139).

Dorsey, Easton, Pa., J. Ind. Eng. Chem., 11, 1919, (284).

'McBain, London, Phil. Mag., 18, 1909, (916).

REAL HYPERSURFACES CONTAINED IN ABELIAN VARIETIES

...

BY S. LEFSCHETZ

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS

Communicated by E. H. Moore, April 29, 1919

...

1. In a recent note of these PROCEEDINGS (April, 1919), I showed that an abelian variety of genus and rank one, V, is birationally transformable into a real one if and only if it possesses 2 p independent linear cycles 71, 72, Y2p, with respect to which p integrals of the first kind have a period matrix of type = Wh,1, wh.p; iwh‚p+1, . . . iwn,2p || ; (h = 1, 2, 1, 2, . . . p), the (w)'s being real. I propose now to investigate the number p' of algebraically distinct real hypersurfaces which V, if real, may have. This number p'≤p, Picard number of V, may also be defined as the maximum number of real hypersurfaces which cannot be logarithmic singularities of a simple integral of the third kind.

[ocr errors]

2. In a general way V, be an abelian variety of rank one, real or not, with the independent linear cycles 71, 72, Y2p. By associating Yμ with Y, we obtain a superficial cycle (u, v) and any other depends upon those of this type. In particular denoting by (A-1) the two dimensional cycle formed by A-1, curve of intersection of 1 algebraic hypersurfaces of the same continuous system as a given one A, we have

20

p

[blocks in formation]

It may be shown that if no integral of the first kind is constant on A the alternate form

[blocks in formation]

is a principal form of as defined by Scorza (Palermo Rendic., 1916), and conversely to a principal form (2) corresponds an algebraic hypersurface A. Moreover to algebraically distinct hyersurfaces correspond linearly independent principal forms from which follows at once p = 1+k, where k is Scorza's index of singularity for .

3. Let us now assume V, real. A real hypersurface A of V, is transformed into itself by T, transformation of the variety which permutes its pairs of conjugate points and this property is characteristic for A. It may be shown that there are real curves A-1,-let the one of No. 2 be one of them, and a its real line (locus of its real points). A small oriented circuit tangent to a in (A-1) is transformed by T into one of opposite orientation, for in the neighborhood of a, T behaves like an ordinary plane symmetry. It follows that T transforms the superficial cycle (4-1) into its opposite. Taking into account the fact that this cycle is a two sided manifold and also

the effect of T upon the linear cycles yμ of No. 1, we find at once that all the m's not of the type mu+, (u, v) are equal to zero, hence p' is equal to the number of independent forms of type.

[merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

This is to be contrasted with Scorza's result 1+k≤ 2p - 1, or p ≤ 2p-1

if is pure.

4. Assuming p'

=

of two forms (3).

2 let L, L', be the matrices formed by the determinants They are both of type

[blocks in formation]

where each square represents a matrix with p rows and columns, the matrices in the main diagonal having only zeroes for terms. As L-1 L' is of the form

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small]

the (A)'s being necessarily real as they can be replaced by their conjugates. Finally the characteristic equation of this complex multiplication

[merged small][merged small][subsumed][ocr errors][merged small][merged small][merged small][merged small]

is necessarily reducible and a perfect square if V, is pure.

5. Let us examine the case of a real hyperelliptic surface of rank one. The number p' has then the value 1 or 2, if the surface is pure not elliptic. A fundamental period matrix corresponding to linear cycles forming a minimum base may be reduced to the form

[blocks in formation]

2,.

where m, n, r, d, are positive integers and ac-b2>0. If y1, 72, are the corresponding linear cycles those of No. 1 are given by

=

γι = γί, γ2

Y2 = 72, 73 =

2√3 — mví — nd√2, Y4 = 2v4 — nv≤ — rôv2,

and in general p = p' = 1, unless there is a singular relation as defined by G. Humbert. If the surface is not elliptic this relation can only be of type

[blocks in formation]

condition of existence becomes now, assuming as we may, > 0,

[blocks in formation]

which assures us of the effective existence of the surface. If there are two singular relations such as (5) the surface is elliptic and p = = p' = 3.

In addition to (5) there may be in the non-elliptic case as well as in the other a singular relation independent of (5) and reducible to the form λ (b2 — ac) + μ = 0, (λ, μ, positive integers)

p'

=

and then p 1, both cases being realizable. Thus there are six distinct types of real hyperelliptic surfaces for which p, p' have the values: (1, 1), (1, 2), (2, 2), (2, 3), (3, 3), (3, 4), the last three corresponding to elliptic cases.

NATIONAL RESEARCH COUNCIL

EXTRACTS FROM THE MINUTES OF THE MEETING OF THE
EXECUTIVE BOARD

AT THE NATIONal Research CoUNCIL BUILDING, April 15, 1919, at 9.30 a.m.

Present: Messrs. Bancroft, Clevenger, Cross, Dunn, Flinn, Hale, Howe, Hussey, Johnston, Leuschner, Merriam, Millikan, Noyes, Walcott, Washburn and Yerkes. Mr. Hale in the chair.

The minutes of the meeting of the Executive Board, March 11 and of the meetings of the Interim Committee, March 18, 25, April 1, and April 8, were approved. The actions taken by the Interim Committee are included in the record of these minutes.

At these various meetings reports were presented by the Chairman of the Council and by the Chairmen of the Divisions of Science and Technology on the progress of the nominations from scientific and technical societies for representatives to serve on the Divisions of Science and Technology of the Council, on the organization meetings of various Divisions and on the election of officers, executive committees, and sub-committees of the Divisions. In accordance with action taken by the Executive Board, each divisional organization will be printed in the PROCEEDINGS after its completion and subsequent approval by the Executive Board, except that sub-committees of Divisions will be reported in the Minutes of the meeting at which they are approved. The membership and officers of the Divisions of Physical Sciences, of Engineering, of Chemistry and Chemical Technology, of Geology and Geography, and of Biology and Agriculture are announced in the present number.

Moved: That the first meeting of each of the permanent Divisions, at which the Division is organized and elects its officers and executive committee, be recognized as the annual meeting of the Division for this year. (Adopted.)

Moved: That all appointments made hereafter shall be considered to terminate on the last day of June of the appropriate year. (Adopted.)

Moved: That the question as to when the present membership of the Council ceases be referred to the Committee on Orzanization with the request that they formulate and circulate resolutions regarding this matter before the next meeting of the Interim Committee. (Adopted.)

Moved: That the Secretary be requested to bring to the attention of the Divisions the provisions of Section 6, Article V, for fixing by lot the term of office of the new members of the Divisions. (Adopted.)

Moved: That the Division of Physical Sciences be authorized to appoint a special committee to consider the advisability of enlarging the functions of the International Bureau of Weights and Measures and to prepare suitable recommendations in this respect.

Moved: That an additional allotment of $1,000 be made to the general maintenance fund of the Engineering Division; and that the transportation expenses of the members of the Division attending the first meeting, be met by the Council. (Adopted.)

« iepriekšējāTurpināt »