The m matrices E1, E2, ... Em thus obtained are linearly independent, as are also the m matrices E'1, E'2, ... E'm; and the former have the same multiplication table as the units of the system (e1, e2, ... ¤m), while the latter have the same multiplication table as the units of the reciprocal system. We now have

€m+1,1);

u=1 v=1

m m

A' = a;e= Σ ai Σ Σ Viuv Eur + €m+1,i

SA',

m

m+1

SA.

m

Σ Σ Yuiu =

m

m

1

m

i=1 u=1

The fundamental properties of the scalar functions given in theorem I are more readily proved for the special case in which the number system is a quadrate than in the general case. What precedes in this section indicates how the properties of these functions may be made to depend upon the properties of the single scalar function of a quadrate.

CLARK UNIVERSITY,

WORCESTER, MASS.

Proceedings of the American Academy of Arts and Sciences.

VOL. XLVIII. No. 18.-APRIL, 1913.

PRELIMINARY STUDY OF THE SALINITY OF SEAWATER IN THE BERMUDAS.

BY KENNETH L. MARK.

IN THE BERMUDAS.1

BY KENNETH L. MARK.

Presented by E. L. Mark, January 8, 1913. Received February 3, 1913.

The objects of this investigation of the salinity and of the temperature of the waters in and about the Bermudas were the collection of data which would supplement those recorded for other parts of the Atlantic Ocean, especially by the "Conseil Permanent International pour L'Exploration de la Mer," and the study of the relation of the salinity to the depth below the surface, to the depth of the sea, and to the locality. A knowledge of these relations was desired as a part of the basis for studies on the distribution of oceanic organisms at the Bermudas.

For these purposes, therefore, samples of water were collected at various places and depths and the temperature of the water was noted in each case. The salinity of these samples was determined by the method used by the "Conseil International." This consists of the complete precipitation of the halides of the sea-water by the requisite amount of a standard solution of silver nitrate. The salinity and density of the samples are then calculated from the analytical results by the aid of the Hydrographical Tables of Knudsen.

Procedure.

The water was collected in a Buchanan2-Nansen3 stop-cock waterbottle, as modified by Dr. H. B. Bigelow, which allows the free passage of water through it during its descent, but can be made to enclose a sample of water at any desired depth. The water was immediately transferred through a brass cock to glass bottles. Care was taken to allow as little evaporation as possible during this transfer. The glass bottles were provided with porcelain stoppers with rubber rings, held on by wire, like the old-fashioned beer-bottle stop

1 Contributions from the Bermuda Biological Station for Research. No. 25. 2 Challenger Report, Narrative, Vol. I, Part 1, p. 112-117.

3 The Norwegian Sea, its Physical Oceanography based upon the Norwegian Researches 1900-1904, by B. Helland-Hansen and F. Nansen. Christiania 1909, in Report on Norwegian Fishery and Marine-Investigations, Vol. II, 1909, No. 2, p. 55.

4 Dr. Bigelow's modification consists chiefly in the substitution of a messenger for the propeller used by Nansen, and will be described in a forthcoming report to be published in the Bull. Mus. Comp. Zoöl., Cambridge Mass.