Lapas attēli
PDF
ePub

teen minutes. Similar studies made at the end of the exposure to environmental temperatures varying from 14° to 30°C. show a distinct tendency towards a flattening out of the curves at the higher temperatures. Thus when the thermal junction was passed down the front and back of the body over exactly the same lines and at the end of two and one-half hours' exposure, the difference between the highest and lowest points in the curves was 10.6°C. with an environmental temperature of 14.6°C.; 9.8°C. with a temperature of 19°C.; 5.4°C..with a temperature of 25.8°C.; and 4.2°C. with a temperature of 30°C.

This study of the temperature of the skin has two important bearings upon all investigations on the heat production of the human body. First, in all researches on direct calorimetry it has been necessary to corrct the heat actually measured by the calorimeter for the heat gained or lost from the body as the result of changes in temperature. Heretofore it has been assumed that as temperature curves measured either deep in the body trunk or in the artificial cavities are similar, a change of 0.1° in rectal temperature indicates a change of 0.1° in the temperature of the entire body. Our observations, particularly with cold temperature environments, show skin temperatures falling several degrees even when interior body-trunk temperatures may be simultaneously rising slightly. The correction of direct heat measurements by records of the rectal temperature is thus open to grave criticism. Unfortunately no substitute correction can as yet be offered. Secondly, these pronounced differences in skin temperature have great significance in any consideration of the socalled 'law of surface area.' It is still maintained by many physiologists that, practically independent of species, the heat production of the quiet organism is determined by its surface area. Our observations show clearly that, contrary to popular impression, the temperature of the skin, presumably one of the most important factors affecting heat loss, is very far from uniform for we have seen that even with well-clothed individuals differences in the temperature of various localities of 4° to 5°C. are of regular occurrence. These observations bring to light and establish one more serious objection to the legality of the conceptions underlying the 'body-surface law.'

ON A CERTAIN CLASS OF RATIONAL RULED SURFACES

BY ARNOLD EMCH

DEPARTMENT OF MATHEMATICS: UNIVERSITY OF ILLINOIS

Communicated by R. S. Woodward, May 19, 1919

As is well known, ruled surfaces, or scrolls as Cayley calls them, may be generated or defined in a number of ways. There exists, for example, a oneto-one correspondence between ruled surfaces and a certain class of partial differential equations, so that the theories of the two classes are abstractly identical.

A much favored method, especially in descriptive geometry, consists in considering ruled surfaces as continuous sets of straight lines, or generatrices, which intersect three fixed curves, the directrices, simultaneously. If these are algebraic curves of orders l, m, n, with no common points, the ruled surface which they determine is, in general, of order 2 l. m. n.

Frequently, ruled surfaces are also defined as systems of elements, either common to two rectilinear congruences, or to three rectilinear complexes.

Of great importance for the following investigation is the definition of ruled surfaces as systems of lines which join corresponding points of an (a, ß) — correspondence between the points of two algebraic curves Cm and C, of orders m and n. If these curves are plane, and if to a point of Cm correspond a points on C, and to a point of C, B points of Cm, then the order of the surface is in general am + Bn.

Finally there is the cinematic method in which ruled surfaces are generated by the continuous movement of the generatrix according to some definite cinematical law. In this connection the description of the hyperboloid of revolution of one sheet is well known.

The literature seems to contain but little about this method of generating ruled surfaces. A number of treatises on differential geometry contain chapters on cinematically generated surfaces.

The class of surfaces here considered is obtained as follows: Given a directrix circle C2 and a directrix line C1, which passes through the center of C2 at right angles to the plane of C2. The generatrix g moves in such a manner that a fixed point M of g moves uniformly along C2, while g in every position passes through C1. The plane e through C1 in which g lies evidently rotates about C1 with the same velocity ke as M. In this plane e, g rotates about M with a uniform velocity kub. When μ = p/q is a rational fraction, the surface generated is also rational and belongs to the class of ruled surfaces generated by means of an (a, B) correspondence between C1 and C2.

When C1 coincides with the z-axis, so that C2 lies in the xy-plane, and we denote by p the distance of the projection P' of a point P on the generatrix g from the origin and by the angle P'OX, the equations of the surface expressed by the parameters p and are

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

It is shown that these may be expressed rationally by p and another parameter t. Also the implicite cartesian equation of the surface is obtained, as well as are the parametric equations of the double curve of the surface. The following theorems are of interest:

Theorem 1. The surface of the class is rational and of order 2 (p + q) or p + q, according as q is odd or even.

Theorem 2. When q is odd the generatrices of the surface cut C1 and C2 in two

point sets which are in a (q, 2 p) - correspondence. C1 and C2 are 2 p-fold and q-fold curves of the surface. The surface has moreover p real and 2 pq − 2 p − q+1 imaginary double generatrices.

[blocks in formation]

2s is even the generatrices cut C1 and C2 in two point sets which are correspondence. C1 and C2 are respectively p- and The surface has no real, but ps p s + 1 imag

in an also rational (s, p)

s-fold curves of the surface.

inary double generatrices.

In the whole discussion the assumption is made, of course, that p and q are relatively prime.

Theorem 3. When q is odd the order of each of the (q

1)/2 double curves is 4 p or 2 q according as q 2 p. They are rational and each lies on a surface of revolution of order 4 generated by the rotation of an equilateral hyperbola about the z-axis.

Theorem 4. When q = 2s is even and s odd, there are (s 1)/2 double curves of order 2p or q according as ps, and one double curve of order p or s, according as ps. When s = 2o is even, there are o double curves of order 2 p or q, according as ps.

The intersections of the double curves with a plane through the z-axis may be arranged according to certain cyclic groups whose orders may be easily determined. One interesting fact is that the surfaces of the class in certain species, are applicable among themselves. The following theorems appertain to this fact:

Theorem 5. Surfaces of the class are applicable to each other when their orders are 2 (p + q) and 2 (m p + n q), and the ratio of the radii of their C2's is m/n, with q odd, p and q, m and n, and m and q as relative primes.

Theorem 6. Surfaces of the class of odd order are applicable to each other when their orders are p + q and m p + n q, and the ratio of the radii of their C2's is m/n. Moreover q is even, and p and m are odd.

As the surfaces of even and odd orders are respectively bifacial and unifacial, we have

Theorem 7. Bifacial and unifacial surfaces of the class are applicable to surfaces of the same type only.

The intersection of a torus with the surfaces of the class yields all so-called cycloharmonic curves. Also the 'bands of Moebius' may readily be cut out from the surfaces.

Among the class here considered are cubic, quartic, and quintic scrolls investigated by Cremona, Cayley, Schwarz and others. Models of these have been and are being constructed in the mathematical model shop of the University of Illinois.

SEP 25 1919

[graphic]

Price per annum, $5.00

peciasisoga

THE PROCEEDINGS is the official organ of the Academy for the publication of brief accounts of important current researches of members of the Academy and of other American investigators, and for reports on the meetings and other activities of the Academy. Publication in the Proceedings will supplement that in journals devoted to the special branches of science. The Proceedings will aim especially to secure prompt publication of original announcements of discoveries and wide circulation of the results of American research among investigators in other countries and in all branches of science. ARTICLES should be brief, not to exceed 2500 words or 6 printed pages, although under certain conditions longer articles may be published. Technical details of the work and long tables of data should be reserved for publication in special journals. But authors should be precise in making clear the new results and should give some record of the methods and data upon which they are based. The viewpoint should be comprehensive in giving the relation of the paper to previous publications of the author or of others and in exhibiting where practicable, the significance of the work for other branches of science.

MANUSCRIPTS should be prepared with a current number of the Proceedings as a model in matters of form, and should be typewritten in duplicate with double spacing, the author retaining one copy. Illustrations should be confined to text-figures of simple character, though more elaborate illustrations may be allowed in special instances to authors willing to pay for their preparation and insertion. Particular attention should be given to arranging tabular matter in a simple and concise manner.

REFERENCES to literature, numbered consecutively, will be placed at the end of the article and short footnotes should be avoided. It is suggested that references to periodicals be furnished in some detail and in general in accordance with the standard adopted for the Subject Catalogue of the International Catalogue of Scientific Literature, viz., name of author, with initials following (ordinarily omitting title of paper), abbreviated name of Journal, with place of publication, series (if any), volume, year, inclusive pages. For example: Montgomery, T. H., J. Morph., Boston, 22, 1911, (731-815); or, Wheeler, W. M., Königsburg, Schr. physik, Ges., 55, 1914, (1-142).

PAPERS by members of the Academy may be sent to Edwin Bidwell Wilson, Managing Editor, Mass. Institute of Technology, Cambridge, Mass. Papers by non-members should be submitted through some member.

PROOF will not ordinarily be sent; if an author asks for proof, it will be sent with the understanding that charges for his corrections shall be billed to him. Authors are therefore requested to make final revisions on the typewritten manuscripts. The editors cannot undertake to do more than correct obvious minor errors.

REPRINTS should be ordered at the time of submission of manuscript. They will be furnished to authors at cost, approximately as follows:

[blocks in formation]

(The charge for fewer than 100 copies is practically the same as for 100.)

Copyright, 1919, by the National Academy of Sciences

« iepriekšējāTurpināt »