peropithecus, in others the crown is much less worn and directly comparable with the relatively unworn premolar crowns of Prosthennops serus (a well-preserved palate of which had been discovered in an earlier expedition), while still others reveal more or ess intermediate conditions. Moreover, the lower teeth which are apparently associated with these apper premolars are unquestionably the same or nearly the same as the corresponding lower teeth of Prosthennops. The still weak link in the chain of evidence consists in the fact that in Prosthennops the premolars that approach the type tooth of Hesperopithecus haroldcookii have two inner roots, whereas the type tooth has a single broad root.

This apparent difficulty may perhaps be met by the hypothesis that the type specimen is a second upper premolar, a tooth which in Prosthennops serus has only a single root; on the other hand, the type is far larger than any known Prosthennops. This much may be said: Nearly every conspicuous character of the type can be matched in one or another of the Prosthennops teeth. Thus, the concave wearing surface of the type is closely approximated in a certain. worn upper molar of Prosthennops; the sharp endng of the enamel on the neck is seen also in the same specimen; the form and direction of the roots are losely paralleled in a third. Another upper molar (found by Professor Abel) and identified by him as Hesperopithecus, in the light of later finds is demontrably Prosthennops.

It is hoped that further exploration this summer (1927) will secure sufficient material to remove all loubt in this matter.

POSTSCRIPT

Last summer (1927) Mr. Thomson made further xcavations in the exact locality where the type of Hesperopithecus haroldcookii was discovered. A rumber of scattered upper and lower premolar and nolar teeth were found in different spots, but every One of them appears to me to pertain to Prosthennops, and some of these also resemble the type of Hesperothecus, except that the crown is less worn.

Thus it seems to me far more probable that we were formerly deceived by the resemblances of the much Forn type to equally worn chimpanzee molars than hat the type is really a unique token of the presence of anthropoids in North America.

WILLIAM K. GREGORY AMERICAN MUSEUM OF NATURAL HISTORY

A NEW THEORY OF POLYGENIC (OR NON-MONOGENIC) FUNCTIONS

If we consider an independent complex variable z=x+ iy

nd a dependent complex function.

w= (x, y) + ip(x, y)

Aw

Az

then in general the limit of the increment-ratio depends not only on the point (x, y) but also on the direction or slope m. The function is called monogenic in the classic case where the limit is independent of m, so that it has only one value at a point. I have proposed recently (in my lectures at Columbia University, and in communications to the National Academy and to the American Mathematical Society) the new term polygenic to describe the case where the limit has many values at a point, one for each slope. Thus for a polygenic function the derivative is not a function of (x, y) or z, but of x, y, m. We write therefore the derivative in the form

(In the special case where the function w is monogenic the circles of course all shrink to points, since in virtue of the Cauchy-Riemann equations h and k vanish so that the radius R is zero.)

To the co2points of the first plane correspond ∞2 circles (in general distinct), that is, a congruence of circles. We call this the derivative circular congruence of the given polygenic function.

Thus while the transformation from the first plane to the second plane is a point transformation, the passage from the first to the third plane gives rise to a contact transformation.

Many noteworthy classes of polygenic functions are obtained by specializing the congruence. Thus if the congruence degenerates into the co1 circles with the center at the origin, the function is of the form

w= f(x-iy)

that is an analytic function (power series) of the conjugate complex variable. If the circles all go through the origin, the components and are dependent, that is the Jacobian must vanish. If the centers all lie on the axis of a, then we obtain the special form

w=W1+iWy,

where W is an arbitrary function of x and y. And

so on.

Returning now to the general theory, we state this fundamental and easily proved theorem:

As the direction or slope m varies at a given point of the first plane, the corresponding point y moves on the derivative circle in the third plane so that its angular rate is always twice that of m and in the opposite sense.

dw

dz

Therefore the complete picture of the derivative

is not a congruence of circles but a congruence

of clocks. Here I use the word clock to denote a circle with a particular distinguished radius vector. We select this to correspond to the direction m=0 at the point in the z- plane. Thus a clock is completely determined by two vectors, namely the central vector H+ iK and the phase vector h+ik.

From the above theorem it follows that there are just three directions m which are parallel to the corresponding radii of the derivative circle, and that these radii are spaced at intervals of 120°. Since this is true at any point, we obtain by integration a triple family of curves (which we call the equiangular family) in the first and third planes.

We next define the mean derivative of a polygenic function as the mean value

where D and D, denote partial differentiation. We thus obtain easily positive and negative powers of this operator.

The mean derivative of a monogenic function is of course a monogenic function. The converse however is not true.

The mean derivative of a polygenic function is sometimes monogenic. This occurs when and only when and obey Laplace's equation, that is, when ❤ and are any harmonic functions.

For this type of harmonic polygenic function, the transformation from the point x+iy to the point H+ iK, which we call the induced center transformation and denote by T', is conformal (direct), though the transformation T from x+iy to u+ iv is in general not conformal. We shall call T in this case a general harmonic transformation. This class of

transformations, which does not form a group, in cludes the total conformal group (made up of directi and reverse conformal transformations) as a special

case.

Further developments of the general theory will be published in the Proceedings of the National Academy of Sciences, the Comptes Rendus, and the Transactions of the American Mathematical Society EDWARD KASNER

COLUMBIA UNIVERSITY, NEW YORK

SCIENTIFIC EVENTS

REPORT OF THE PRESIDENT OF THE
CARNEGIE INSTITUTION

THE trustees of Carnegie Institution of Washington met in annual session on December 9, Elihu Root presiding. In recognition of the fact that the institution is completing a quarter century of activity, President Merriam, in his formal report covering the work of the institution for 1926-27, briefly characterized the poli cies of the quarter century.

He said that in the first years the institution's grants were commonly made for specific projects to run for limited periods. These covered a relatively wide range of subjects, affording an important stimulus to many types of agencies. In later years the tendency devel oped to center upon major projects which required sustained effort and concentration of funds. Thi tendency resulted in the development of department in the institution's organization, each devoted to it specific subject and under leadership of an investi gator of exceptional vision and ability. Although th practice of giving minor grants to distinguished ind viduals for special projects was continued, in man cases advantage was found in relating such problem to that department of the institution best fitted to c operate. Still more recently a relation between d partments has developed comparable to that which ha developed in some instances between departments a individual investigators.

President Merriam summed up his observations the institution's policy as it has evolved during t quarter century by saying:

The institution to-day contains all the elements th have arisen in the course of study of its problem. The are still widely distributed special grants. The great departmental activities still represent concentrated eff in specific fields. The increasing mutual support has diminished initiative of the individual or of the gro but it has added an element which with the passing time becomes more and more valuable, both in effort concentrate upon special projects and in keeping t view of the larger field so desirable in long-contin researches.

Plans involving an expanded program of activities in the fields of early American cultures and of plant biology were also referred to by the president.

In respect to the first of these he said that the specific investigations heretofore undertaken by the institution in American archeology had been initiated in the hope that they might lead to some suggestion of laws which have governed in the development of the varying types of early peoples and cultures in America. He pointed out that the time had now come in the work of the Carnegie Institution in this field when the results should be interpreted in the light of what has been learned about the American problem in other regions and by other agencies.

The president also stated that Dr. Alfred V. Kidder, who has worked with distinction in early American history and is deeply interested in the wider aspects of the matter, has accepted leadership for the institution in its broadened activities in this field of research. This change implies no lessening of interest in the investigation of the Maya culture of Yucatan, for the results which the institution has obtained, he assured the trustees, amply justify adherence to its fundamental plan of furthering historical studies in Middle America.

In referring to proposed changes in the plant biology program the president said that during the past two decades the institution had attempted to advance the boundaries of knowledge by intensive effort at several critical points along the margin of the field. Such, he suggested, have been the researches in problems of life process in plant physiology, on the relation of life development to special types of conditions as in work at the Desert Laboratory, on the relation of life progress to environment, on problems of plant heredity, on questions touching relation of classification to heredity and to influence of environment and on problems presented by the history of plant life during the ages.

In 1926 a small committee of the institution's leading investigators in this field was asked to formulate a program for future guidance after full consideration of the matter. The report of this committee, the trustees were informed, indicates that a greater unity of attack would be profitable to all, without real loss or hardship and that a larger measure of unity in administration would facilitate the entire plant biology program.

President Merriam also spoke of the opportunity afforded the institution's investigators during the year of participating in international meetings. He believes that such meetings and the contacts with workers in related fields which they provide have distinct value. On this point he said:

Through these relationships there has developed both the accumulation of materials arising from studies by other institutions and the contribution from our own researches, which are thus subjected to constructive review by experts in related fields. It is believed that such extension of our relationships is one of the important means to be used in finding how our own course should be steered in the unknown fields toward which we are always moving.

Dr. Merriam presented to the trustees a report covering in detail the progress of research carried on by the institution during 1926-27. This will soon be available for distribution in the form of Year Book, No. 26.

The trustees upon completing the business of the day spent the afternoon with their friends in viewing the exhibition prepared by members of the institution staff. This exhibition, which is set up at this time every year at the Administration Building, Washington, D. C., is designed to show the progress made in significant research activities. For three days following the annual meeting of the board of trustees, December 10, 11 and 12, the public generally was invited to view the exhibits.

EXPLORATIONS IN ALASKA BY THE U. S. GEOLOGICAL SURVEY

THE U. S. Interior Department announces the completion of the field work of another exploratory expedition in Alaska by the Geological Survey and the bringing back of maps and information regarding a tract of more than 2,000 square miles in the Alaska Range and adjacent country on the west side of Cook Inlet, in the environs of Mount Spurr, that has hitherto been shown as a blank area on all authoritative maps. This exploration is one of the series that the Geological Survey has been making throughout the last thirty years. The party consisted of S. R. Capps, geologist in charge; R. H. Sargent, topographic engineer, and four camp men. Transportation in the field of the necessary provisions, supplies and equipment for 100 days was furnished by a pack train of fifteen horses. From the time when the party landed at Trading Bay, on the west side of Cook Inlet, about the middle of June, until it returned to that place at the end of the field season, about the middle of September, the members were entirely out of communication with the rest of the world.

Among the many items reported are the discovery and mapping of a large river, numerous lakes, glaciers and mountains and an active volcano. The newly discovered large river is the Chakachatna, whose drainage basin covers an area of more than 1,100 square miles. This stream is a roaring torrent far too swift and too deep to be forded even with horses; in fact, measure

ments of its current at several places showed that it was flowing at an average speed of 15 miles an hour. This river rises in a superb lake, Lake Chakachamna, 23 miles long, which is hemmed in between lofty mountains and impounded behind a great glacier that lies athwart the general trend of the valley. The distribution of the rivers that head against the Chakachatna can now be predicted with considerable assurance. Thus, to the south are rivers that probably flow in part into Lake Clark; to the west and northwest are tributaries of Stony River and of the South Fork of Kuskokwim River.

Many of the mountain peaks are ragged pinnacles which could be climbed, if at all, only with great difficulty. The highest peak of the region is Mount Spurr, which rises to an altitude of 11,000 feet and is clearly visible from the coast. Although the flanks of this mountain are in large part covered with perpetual snow and glaciers, the mountain was found to be an old volcano that is still active. When the weather was favorable a plume of steam could be seen rising from a point near its crest to a height of more than a thousand feet. This discovery therefore establishes Mount Spurr as the most northerly of the long series of known active volcanoes that occur at intervals along the west coast of Cook Inlet and extend westward into the Alaska Peninsula and Aleutian Islands.

ENDOWMENT FOR THE ARNOLD

ARBORETUM

FRIENDS of the late Professor Charles Sprague Sargent and others interested in the Arnold Arboretum of Harvard University, in Jamaica Plain, on November 30 gathered at Sherry's Restaurant, New York City, to hear of plans for the completion of a $1,000,000 endowment fund for the arboretum. The fund, about $600,000 of which has been raised since Professor Sargent's death last March, will be used to perpetuate the great botanical station as he had planned it. About 500 persons attended the dinner, held under the auspices of the New York committee.

David Fairchild, agricultural explorer in charge of foreign plant introduction for the U. S. Department of Agriculture; Professor Oakes Ames, present supervisor of the arboretum and successor to Professor Sargent, and the Right Reverend William Lawrence, formerly Bishop of the Episcopal Diocese of Massachusetts, were the speakers. All urged support of the arboretum, painting it as a vital part of the country's life and as a great power both in the economic world and for the beauty of the nation.

It was announced on December 3, by Mr. J. P. Morgan, who is acting as treasurer, that since the opening of the New York campaign to raise part of the proposed $1,000,000 endowment fund for the ar

boretum, $109,250 has been contributed by Ne Yorkers.

A gift of $50,000 to the fund has been offered b Edward S. Harkness, on condition that the New Yor committee match the $460,000 already contributed i Boston.

New York contributors to the fund so far ar Moreau Delano, $25,000; J. P. Morgan, $20,000; Mr Harold I. Pratt, $15,000; Mrs. W. Bayard Cutting $5,000; William Adams Delano, $5,000; T. A. Have meyer, $5,000; Mrs. James H. Metcalf, $5,000; Charle A. Stone, $5,000; William Nelson Cromwell, $2,500 Henry W. de Forest, $3,000; John E. Aldred, $2,500 R. W. de Forest, $2,500 plus $500; J. N. Jarvi $2,500; anonymous, $1,000; Paul D. Cravath, $1,000 Mrs. Max Farrand, $1,000; Mrs. W. L. Harknes $1,000; Clarence Hay, $1,000; Charles Hayde $1,000; G. O. Muhlfeld, $1,000; Anton G. Hodenpy $500; Mrs. Arthur Curtiss James, $500; Mr. and Mr A. R. Graustein, $500; Victor Morawetz, $500; W. H Bush, $250; O. M. Eidlitz, $250, and Elihu Root, $25

THE FIRST MEETING OF THE COLORADO WYOMING ACADEMY OF SCIENCE

THE first meeting of the Colorado-Wyoming Acad emy of Science was held on November 25 and 26, a the University of Wyoming. The new organizatio has an initial enrolment of 166 members chiefly from the eight educational institutions of these two state

Sectional meetings were held in the new engineerin building of the University of Wyoming and forty one papers were presented on chemistry, physic botany, zoology, geology and social science. At th business meeting on the afternoon of the second da the constitution was formally adopted and the follow ing officers elected:

Dr. Aven Nelson, University of Wyoming, President. Dr. O. C. Lester, University of Colorado, Vice-presiden Dr. L. W. Durrell, Colorado Agricultural College, Se retary.

Dr. E. B. Renaud, University of Denver, Treasurer.

Closing the session a banquet was given the mem bers by the University of Wyoming.

SCIENTIFIC NOTES AND NEWS

THOMAS ALVA EDISON has been elected an honorar member of the British Institution of Electrical Engi

neers.

PRESENTATION of the Chandler medal by Columbi University will be made to Professor Moses Gomber on December 15, when he delivers the Chandler le ture on "Free Radicals in Chemistry-Past an Present."

DR. WILLIAM J. MAYO, of Rochester, has been named a commander of the Royal Order of the North Star by King Gustav of Sweden.

PRINCE ALBERT DE LIGNE, ambassador to the United States from Belgium, conferred on Dr. Chevalier Jackson, chief of the bronchoscopic clinic at Jefferson Hospital, the Knighthood of the Order of Leopold, on November 22, at an informal luncheon in the Belgian Embassy, Washington, D. C.

Ar the meeting of the British Chemical Society on November 8, Sir Joseph J. Thomson was elected an honorary fellow of the society.

THE Russian expert commission for the awarding of the Lenin prizes for scientific work has made awards to Professor A. N. Bach for work in the province of biological chemistry; to Professor V. P. Vorobyov, Kharkov, for work in anatomy; to Professor K. K. Gedroits for work in agricultural chemistry and soil science, and the late Professor L. A. Chugaye for work in the affinity of precious metals. THE James Scott prize of the Royal Society of Edinburgh, for the period 1922-1926, "for a lecture. or essay on the fundamental concepts of natural philosophy," has been awarded to Sir Joseph Larmor.

THE Royal Meteorological Society has awarded the Symons memorial gold medal for 1928 to Professor Hugo Hergesell, director of the Aeronautical Observatory, Lindenberg, for distinguished work in connection with meteorological science. The medal, which is awarded biennially, will be presented at the annual general meeting on January 18.

ALEX Dow, of Detroit, has been elected president of the American Society of Mechanical Engineers, succeeding Charles M. Schwab, chairman of the Bethlehem Steel Corporation. Vice-presidents are John H. Lawrence, New York; E. A. Muller, Cincinnati; Newell Sanders, Chattanooga, Tenn.; Paul Wright, Birmingham, Ala. The managers named are L. B. McMillan, New York; William A. Hanley, Indianapolis, Ind.; F. H. Dormer, Milwaukee, Wis.

SIR CHARLES CLOSE, formerly director-general of the British Ordnance Survey, has been appointed by the council of the Royal Geographical Society president of the society until the anniversary meeting next June in succession to Dr. D. G. Hogarth, who died on November 6.

DR. ROBERT F. MEHL has been appointed superintendent of the division of physical metallurgy in the Naval Research Laboratory, not director of the laboratory as was incorrectly stated in the last issue of SCIENCE. Captain D. E. Theeleen is director of

the laboratory and Commander E. G. Oberlin is assistant director.

THE University of Manchester has conferred the title of professor emeritus upon Professor R. B. Wild on his retirement from the Leech chair of materia medica and therapeutics.

PROFESSOR SUSLOV, director of the Odessa Polytechnical Institute, has celebrated his seventieth birthday, in which the Odessa scientific circles largely participated.

LUDLOW GRISCOм has been appointed assistant director of the Harvard Museum of Comparative Zoology and has resigned from the assistant curatorship of ornithology at the American Museum of Natural History.

DR. WILBERT W. WEIR, associate soil technologist in charge of editorial work in the U. S. Bureau of Chemistry and Soils, has resigned to accept a position with the Chilean Nitrate of Soda Educational Bureau, New York.

PROFESSOR GROVER D. TURNBOW, head of the division of dairy industry at the University of California, has been given leave of absence to engage in commer

cial work.

AT Duke University, W. H. Hall, professor in charge of engineering, has returned after spending his year of leave at the University of Wisconsin. Harold C. Bird, who has been acting as professor of civil engineering during Professor Hall's absence, continues as professor of civil engineering.

DR. D. C. CARPENTER, associate chemist at the New York State Agricultural Experiment Station, who has been working for the past few years on the structure of the casein molecule, has been invited by Dr. T. Svedberg to make use of the centrifuge in his laboratory at the University of Upsala to ascertain the molecular weight of casein. Dr. Carpenter sailed for Sweden on December 9, under the auspices of the International Education Board, and will spend the coming year with Dr. Svedberg.

PROFESSOR JAMES H. BREASTED, director of the Oriental Institute of the University of Chicago, which has six expeditions in the field in search of records of lost civilizations, left Chicago on December 5 to inspect the work now in progress. Professor Breasted, his son Charles and his secretary, R. J. Barr, sailed from New York on December 10.

DR. P. J. VAN LONKHUIJKEN, director-general of health of the Netherland East Indies, is in the United States inspecting methods of public health and disease. control employed by the U. S. Public Health Service.