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ale; the kwalls, rtically

rather prominent sulcate apex subtended by four somewhat spreading bisulcate prominences. Appendage relatively very large, the subequal cells rounded as in the male. Perithecium, including stalk 140-160X 20-25 μ. Appendage 30 X 20 μ.

μ.

On Laemobothrium atrum from Coot, New England. M. C. Z., No. 1537.

This species is most easily distinguished by its unusually large appendage, which resembles a stout spore of Puccinia. It seems most nearly related to T. Lipeuri, the perithecia being very similar. The mode of growth is however, quite different. The rhizoids are entirely broken off in all the specimens.

Trenomyces circinans nov. sp.

Male individual. Corticating cells few and irregular, producing usually not more than two to four antheridia. Antheridia of the usual form, the body bent often at a right angle to the slender stalkcell or sometimes recurved, the stalk 18 X 4 μ, the body 18 X 14 μ. Appendage relatively small, the cells about equal, 18 X 11 μ, the distal cell blunt pointed.

Female individual. Swollen portion of the rhizoid bearing several horizontal or upcurved lobes from which arise usually furcate smaller lobes running to slender threads of no great length. Perithecia two to four, usually strongly circinate when young, at maturity typically bent or even recurved, rarely straight, the stalk relatively slender, the body often rather abruptly distinguished, broader distally below the tip, which may be subtended by a distinct elevation on one side and is well distinguished, its margin usually slightly convex, separated by a slight constriction from the crown formed by four symmetrically placed somewhat spreading lobes which surround the hardly prominent apex, the whole surface of the stalk and body more or less distinctly roughened or granular, the walls much thickened. Appendage relatively small like that of the male. Perithecium including stalk 225-280 X 28-35 μ; the stalk 70-125 X 10 μ or broader. Appendage 20 X 10-14 μ.

On various parts, especially the head of Lipeurus sp., on pigeons, Kingston, R. I., No. 1549; on L. baculus, Elbing, Prussia (Dr. Müller); on Docophorus Californicus, California, No. 1555 (Kellogg No. 666); on D. Montereyi, No. 1554 (Kellogg No. 264c).

The Californian forms on Docophorus are not quite so well marked as those from Prussia and Rhode Island which, by their abruptly

curved habit, slender stalks, and roughened surface, are clearly distinguished from other species of the genus. The tip of the perithecium in well developed specimens is not unlike that of Arthrorhynchus Eucampsipodae, but the conformation varies considerably and comparatively few specimens have a well defined subterminal hunch. Several specimens on Docophorus colymbinus, Nos. 1556-7 (Kellogg, Nos. 14a and 12a), differ distinctly in that the tip is unmodified and hardly distinguished, the stalks stouter and less well distinguished. Further material may indicate that this form is distinct.

Trenomyces gibbus nov. sp.

Male individual unknown.

Female individual. General structure like that of T. histophorus. Swollen portion of the rhizoid producing several, horizontal lobes. Corticating cells very irregular, completely concealing the somewhat irregular basal cell, giving rise to numerous perithecia. Perithecia faintly tinged with yellowish, stout elongate, the stalk not distinguished from the body, the whole indistinctly roughened, and having the appearance of a goose's neck and head owing to a subterminal protrusion which causes the tip and apex to be bent to one side at an angle 45° or more; the tip nearly symmetrical above the protrusion, broadly conical, the apex rather narrow, subtruncate, slightly indented. Total length of perithecium 300 μ, including stalk, which may be 30 μ broad just above its origin; the tip above the hunch, 32 μ long, the base 28 to 30 μ broad, the apex about 7 μ. Appendage 25 X 10 μ. Described from a single female on Lipeurus longipilus. No. 1563 (Kellogg, No. 128d), California.

This form is so peculiar that I have not hesitated to describe it from a single well developed female in good condition. There are a dozen or more perithecia on the specimen in various stages of development, the four which are mature suggesting the heads and necks of a flock of geese. The distal cell of the appendage is somewhat longer than the basal, tapering from base to apex.

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Proceedings of the American Academy of Arts and Sciences.

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THE SPACE-TIME MANIFOLD OF RELATIVITY. THE NON-EUCLIDEAN GEOMETRY OF MECHANICS

AND ELECTROMAGNETICS.

BY EDWIN B. WILSON AND GILBERT N. LEWIS.

THE SPACE-TIME MANIFOLD OF RELATIVITY.

THE NON-EUCLIDEAN GEOMETRY OF MECHANICS AND ELECTROMAGNETICS.

BY EDWIN B. WILSON AND GILBERT N. LEWIS.

Introduction.

1. The concept of space has different meanings to different persons according to their experience in abstract reasoning. On the one hand is the common space, which for the educated person has been formulated in the three dimensional geometry of Euclid. On the other hand the mathematician has become accustomed to extend the concept of space to any manifold of which the properties are completely determined, as in Euclidean geometry, by a system of self-consistent postulates. Most of these highly ingenious geometries cannot be expected to be of service in the discussion of physical phenomena.

Until recently the physicist has found the three dimensional space of Euclid entirely adequate to his needs, and has therefore been inclined to attribute to it a certain reality. It is, however, inconsistent with the philosophic spirit of our time to draw a sharp distinction between that which is real and that which is convenient,1 and it would be dogmatic to assert that no discoveries of physics might render so convenient as to be almost imperative the modification or extension of our present system of geometry. Indeed it seemed to Minkowski that such a change was already necessitated by the facts which led to the formulation of the Principle of Relativity.

2. The possibility of associating three dimensional space and one dimensional time to form a four dimensional manifold has doubtless occurred to many; but as long as space and time were assumed to be wholly independent, such a union seemed purely artificial. The idea of abandoning once for all this assumption of independence, although fore-shadowed in Lorentz's use of local time, was first clearly stated by

1 See, for example, H. Poincaré, La Science et l'Hypothèse.

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