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peratures of 49° C. in the sun in an unventilated glass house were heated further by the use of electric grills. Temperatures were taken by mercurial thermometers with bulbs of the clinical type thrust into joints within a few centimeters of the one being measured, but which had equivalent exposure.1

The elongation of the joints during this youngest stage is directed by the temperature, and the retardations due to maximum night transpiration and acidity which come in later are not yet manifest. The rate of elongation therefore is greatest in midday and early afternoon. Such a joint showing a temperature by the inserted thermometer of 43.5° C. was subjected to the additional heating of the electric grill at 1:30 P.M. At 2 P.M. the temperature passed 51° C. with growth still in progress, the rate but little lessened from that of 1 mm. in 24 hours which it was showing at the beginning of the test. The temperature was now raised slowly until 3 P.M. the joint stood at 51.5° C., the maximum at which growth had ever been observed in any seed plant. At 3:10 a temperature of 54.5° C. was reached and five minutes later the readings were 55.5° C. The joint was kept for an hour between 55° and 55.5° C. during which time the auxograph tracing showed a retardation but not a stoppage of growth. The heat was shut off, the temperature soon falling to 42° C. and to 19° C. at 9 P.M., when the record assumed the character of that of the preceding day of the same joint and of a similar one standing near it.

A repetition of the tests was made next day at 10 A.M. when the joint stood at 33.5° C. The heaters were brought into action, the joint reaching 55° at 10:45 A.M. The preparation stood in the sun and was under normal

1 MacDougal, D. T., and H. A. Spoehr, "Growth and Imbibition," Proc. Amer. Phil. Soc., 56, 289– 352, 1917. McGee, J. M., "The Effect of Position upon the Temperature and Dry Weight of Joints of Opuntia," Carnegie Inst. Wash. Year Book for 1916, p. 73. MacDougal, D. T., "Hydration and Growth," Carnegie Inst. Wash. Pub. 297, 1920. DeVries, H., "Matériaux p. 1. connaisance d. l'influence d. 1 temperature s. 1. plantes," Arch. Néerlandaises, III., p. 3, 1870.

conditions of ventilation and transpiration. Readings of 54.5° C. to 55.5° C. were made for a period of an hour and a half during which period the elongation was 0.2 mm. or near the maximum rate for the species and was still continuing. One heater was removed at 12:15 midday and ten minutes later the joint had fallen to 49.5° C. The cooling had resulted in a minute reverse movement of the auxograph recording lever of a character which could only be attributed to the contraction of the metal and clay of the setting. The temperature of the joints had fallen to 32° C. by 3 P.M. with no noticeable diminution of the rate, the maximum being taken to lie at some point over 40° C.

A comparison of the thermometer with U. S. Bureau of Standards No. 7618 gave an error so small as to be negligible with regard to the above data. Furthermore the young joint continued its growth at a rate normal to its developmental stage.

These and previously published measurements establish the following points:

1. Growth in Opuntia may begin at 9° C. and extend to 55° C.

2. Young joints of Opuntia may endure the maximum of 55° C. observed in mature joints in midsummer, for periods of an hour and a half, resuming elongation at lower temperatures with no perceptible after-effects.

3. A new high record for growth in Opuntia and for the higher plants of 55° C. (131° F.) has been established by these experiments.

4. The maximum rate of growth of Opuntia occurs between 37° C. and about 47° 49° C., under which conditions a biocolloid consisting of 9 parts agar and 1 part protein undergoes maximum swelling in water.2

5. The cell colloids of Opuntia include a large proportion of pentosans or mucilages, the colloidal condition of which is in general less affected by the temperatures used than albuminous substances. It is to be noted however that bacterial cells, which are highly albuminous, may withstand high tempera

2 MacDougal, D. T., "The Relation of Growth and Swelling of Plants and Biocolloids to Temperature," Proc. Soc. Exper. Biol., 15, 48-50, 1917.

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THE two hundred and fourteenth regular meeting of the American Mathematical Society was held at Columbia University, on Saturday, February 26, 1921, extending through the usual morning and afternoon sessions. The attendance included thirty-five members Ex-president H. B. Fine occupied the chair. One hundred and fifteen new members were elected, and twenty-four applications for membership in the society were received.

The council voted to accept the invitation to affiliate with it extended to the society by the American Association for the Advancement of Science. Professor E. B. Van Vleck was appointed representative of the society in the division of physical sciences of the National Research Council, to succeed Professor H. S. White. The final report of the committee on membership and sales was presented by its chairman, Professor E. R. Hedrick; in all one hundred and thirty-two applications for membership have been received through this very efficient committee. Questions having arisen concerning dues of foreign members, concerning sales and exchanges of publications with foreign societies and libraries, and concerning individual or concerted efforts to aid foreign journals, a committee was appointed by the council to consider these and related problems.

A letter was read to the council from ex-secrebary F. N. Cole donating to the society the sum which accompanied the testimonial tendered him at the preceding meeting of the society in recognition of his very distinguished services. It was voted that the council accept the gift and extend to Professor Cole its heartiest appreciation of his generosity; it was further voted that this fund shall constitute, and be designated as, the Cole Fund. A committee was appointed to consider the use to which the income can best be devoted. The council approved the suggestion that the present volume of the society's Bulletin be inscribed to Professor Cole.

A letter of felicitation was sent to Professor Mittag-Leffler, of Stockholm, on the occasion of the seventy-fifth anniversary of his birth.

The following papers were read at this meeting: Coefficient of the general term in the expansion of a product of polynomials: L. H. RICE.

The mathematical theory of proportional representation, with a substitute for least squares: E. V. HUNTINGTON,

On the apportionment of representatives: F. W. OWENS.

On the polar equation of algebraic curves: ARNOLD EMCH.

Generalization of the concept of invariancy derived from a type of correspondence between func tional domains. Second proof of the finiteness of formal binary concomitants modulo p: O. E. GLENN.

Concerning the sum of a countable number of point sets: R. L. MOORE.

On the simplification of the structure of finite continuous groups with more than one two-param eter invariant subgroup: S. D. ZELDIN.

Periodic functions with a multiplication theorem: J. F. RITT.

Note on equal continuity: J. F. RITT. Expressions for the Bernoulli function of order p: I. J. SCHWATT.

The expansion of a continued product: I. J. SCHWATT.

Method for the summation of a family of series: I. J. SCHWATT.

Note on the evaluation of a definite integral: I. J. SCHWATT.

A property of the Pellian equation with some results derived from it: JOHN MCDONNELL.

A necessary and sufficient condition that the sum of two bounded, closed and connected point sets should disconnect a plane: ANNA M. MUL


Some empirical formulas in ballistics: T. H. GRONWALL.

Summation of a double series: T. H. GRONWALL A geometrical characterization of the paths of particles in the gravitational field of a mass at rest: L. P. EISENHART.

The equations of interior ballistics: A. A. BEN


The next meetings of the society will be at Chicago on March 25 and 26, and at New York, in April.

R. G. D. RICHARDSON, Secretary


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WE record, with deep regret at his passing, but with high appreciation of his long and valuable service to astronomical science, the death of Sherburne Wesley Burnham, emeritus professor of practical astronomy at the Yerkes Observatory, of the University of Chicago.

Born on December 12, 1838, in the upper valley of the Connecticut, at Thetford, Vermont, Mr. Burnham had the ordinary advantages of the district school, supplemented by some study in the local academy, but he did not go to college. He became an expert stenographer and shorthand reporter, long before the days of the typewriter, and this was his profession for some thirty years. During the Civil War he served in his professional capacity with the Union Army while it was occupying the city of New Orleans. He came to Chicago, after the close of the war, and became attached to the United States Courts.

His interest in astronomy must have developed very early in the sixties, for he purchased his first telescope during a visit to London in 1861; and in 1870 he became the possessor of a fine six-inch refractor, a masterpiece of Alvan Clark, which he had ordered in 1869. Mr. Burnham's vision was extraordinarily keen, for among the 451 new double stars which he discovered with that instrument many were found by other observers to be extremely difficult to resolve with much larger instruments.

In 1873 and 1874 he sent five lists of new double stars to the Royal Astronomical Society, which were published in the Monthly Notices. At first he had no micrometer, and was obliged to give estimated angles and distances. A correspondence developed with Baron Ercole Dembowski, who gladly made

the micrometric measurements, with his excellent skill, using a refractor of 162 mm. aperture at Gallarate, in Italy. Two lists covering 136 new double stars were printed in the Astronomische Nachrichten in 1875 and 1876. A short list followed in the American Journal of Science in 1877 and in Monthly Notices for the same year. In 1879 his new doubles from Nos. 483 to 733 were published in the forty-fourth volume of the Memoirs of the Royal Astronomical Society, together with micrometric measures of 250 other stars.

During the years from 1877 to 1881 and 1882 to 1884, Mr. Burnham had the use of the splendid 18 inch Clark refractor of the Dearborn Observatory, then set up in the tower attached to the old Chicago University.

In 1879 he was requested by the trustees of the Lick Trust to test the conditions on Mt. Hamilton. He took his 6-inch refractor, now equipped with circles and a driving clock, to Mt. Hamilton and made observations from August 17 to October 16. His highly favorable report settled the choice of Mt. Hamilton as the site for the Lick Observatory. In 1881 he went again to Mt. Hamilton, by request, and observed the transit of Mercury with the 12-inch telescope.

During some six months of 1881 he was astronomer, under E. S. Holden, at the University of Wisconsin, where the 15.5-inch telescope of the Washburn Observatory had recently been erected. While there he discovered and measured 88 new double stars; and he measured a large number of double stars "Selected from his MS. General Catalogue of Double Stars, as specially needing observation." These observations appeared in Vol. I. of the Publications of the Washburn Observatory in 1882. Mr. Burnham's famous 6-inch refractor ultimately become a part of the equipment at Madison.

On the inauguration of the Lick Observatory in 1888, with Professor Holden as director, Mr. Burnham received the appointment as astronomer, and thus had abundant opportunities for the use of the great 36-inch Clark refractor for the continuance of his work. At the Lick Observatory he intro

duced the principle of using the telescope for all it was worth while the sky permitted: in other words, no part of the night when the sky was clear was given up for any bodily weariness of the observer. In 1892, owing to certain conditions at Mt. Hamilton which were unacceptable to Mr. Burnham, he returned to Chicago, where he was offered the highly responsible position of Clerk of the United States Circuit Court. Incidentally he was receiver of the Chicago and Northern Pacific Railroad Company from 1897 to 1902.

Mr. Burnham was in charge of the expedition from Lick Observatory to observe, at Cayenne, the solar eclipse of December 21-22, 1889. Good results were secured, due in no small measure to Mr. Burnham's large experience in photography. The report was written by Burnham and his associate, Mr. Schaeberle, and published in 1891 in a small volume from the Lick Observatory.

On the inauguration of the Yerkes Observatory in 1897, Burnham became an active member of the staff, making his observations throughout the nights of Saturday and Sunday and returning to his duties in the court on Monday morning. In 1902 he resigned his position with the court, despite the life tenure of that office. This gave him more time for his astronomical studies, but he still retained his residence in Chicago, coming to Williams Bay for observations on two nights in the week. He became Professor emeritus in 1914, at the age of 75, the statute of the University of Chicago requiring retirement at 70 having thus far been waived in his case. Although the opportunity for using the 40-inch telescope still remained open to him as before, he hardly availed himself of it, and his last observations here were made on May 13, 1914.

Vol. II. of the Publications of the Lick Observatory contain his observations from August, 1888, to June, 1892, and his fourtenth to nineteenth catalogues of new double stars discovered at the Lick Observatory in that period, including the numbers from B 1026 to ẞ 1274. The search for new doubles was made chiefly with the excellent 12-inch telescope. He also found some new nebulæ,

and measured the positions of numerous planetary nebula which are given in the same volume. His orbits for several of the more interesting systems on which he had been working appear at the end of that volume. It will be seen that Mr. Burnham had largely given up the search for new double stars while at the Lick Observatory, regarding it as more important that accurate observations should be made of the systems already discovered, particularly those for which large instruments were necessary.

Vol. I. of the Publications of the Yerkes Observatory, issued in 1900, is entitled "A General Catalogue of 1290 Double Stars Discovered from 1871 to 1899 by S. W. Burnham." It gives in order of right ascension the history of all of the Burnham stars up to B No. 1290. Aside from his own observations, it summarizes the results of all other observers of these stars and gives diagrams and orbits, by the author and others, of several interesting systems. He did not allow himself to be distracted from his specialty by the allurements of other fields of observation: it was seldom that he looked at nebulæ unless there were double stars to be measured therein; and he had no time for observing comets, however interesting. He made an exception in locating Halley's comet on September 15, 1909, two nights after it had been first caught on a photographic plate by Wolf at Heidelberg: thus Burnham's eye was the first to see the comet, then an extremely faint speck, on this return to perihelion.

During the beginning of Mr. Burnham's use of the 6-inch telescope, he felt the great need of a single catalogue of all double stars in the Northern Hemisphere and he therefore arranged a manuscript catalogue of all known double stars within 121° of the north pole. This was conveniently indexed and proved of great service to the observer. He revised it in two MS. editions, the third of which allowed ample room for expansion and is still in use. The preparation of this catalogue had entailed a great amount of labor, as it was constantly kept up to date. Mr. Burnham

says of it that "very few will fully appreciate the enormous amount of hard work which has been necessarily expended in the preparation of such a work. . . . It should be remarked in this connection that with the exception of the four years from 1898 to 1902 all this astronomical work, with the telescope and otherwise, has been done when eight or more hours of at least six days in the week were very much occupied with other and different affairs of life." After his retirement from active observations, Mr. Burnham turned this MS. catalogue and the responsibility of its up-keep over to Professor Eric Doolittle, whose premature death in 1920 is much lamented. From him, by prior arrangement, this passed on to Professor Robert G. Aitken, of the Lick Observatory, who thus carries on the work which will eventually result in a new edition of the "General Catalogue of All Double Stars," now to be mentioned. Efforts had been made for many years to have this great work published, but it could not be brought about until the Carnegie Institution of Washington in 1905 undertook to publish it. The composition was done with great care by the University of Chicago Press, and Part I. was published before the close of 1906. It lists 13,665 double stars and summarizes the numerical information about them, in a quarto volume of 275 pages. Part II., of 1,086 pages, gives details of all important observations of the pairs, with many diagrams. It constitutes a magnum opus of which any scientist could be justly proud.

With the 40-inch telescope of the Yerkes Observatory, Mr. Burnham gave no time to the discovery of new doubles. In fact, he avoided them, if possible, and occasionally mentioned seeing some which he did not record. In recent years he took a good deal of interest in the determination of the proper motions of the brighter stars by micrometrically connecting them with neighboring faint stars, for which a negligible proper motion could be assumed. This work was largely to lay the foundation for a greatly increased knowledge of proper motion in the future. Mr. Burnham realized very fully the great advan

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