destruction of the wood as is 40 per cent. loss at 60 per cent. or 80 per cent. moisture content.

The maximum moisture content at which decay can take place in any of our commercial woods is about die 190 per cent. or 200 per cent. Beyond that point, the water drives out so much of the air that insufficient is left to support growth of these wood destroy

ers.

The relation of this problem to specific gravity is obvious, of course. Presupposing that a certain definite volume of air is necessary to support the growth of these wood destroyers, the moisture content favoring the maximum amount of decay or inhibiting decay entirely will vary inversely with the specific gravity. The incomplete series show that satisfactorily. With woods of three specific gravities, there have been obtained three points for both the limit of optimum decay and the inhibition point in terms of percentage of water. It can not of course be determined as yet whether these points form a straight line on the graph or are on a parabolic curve. Tests are now in progress not only to fill the gaps existing in the present series, but also to obtain two more points in the graph by growing certain of the fungi upon southern yellow pine of specific gravities .65 and .75. Whether these points will then form a straight line or a curve, it is expected to be able to prove that the durability of some of our heavy woods, like white oak and best southern pine, etc., is not due to tannin, resin or anything more than its high specific gravity-i.e., its small lumina, and hence small amount of air available for fungous growth.

DEPARTMENT OF BOTANY, BROWN UNIVERSITY,

PROVIDENCE, R. I.

WALTER H. SNELL, NATHANIEL O. HOWARD, MYRON U. LAMB

THE AMERICAN MATHEMATICAL

SOCIETY

THE thirty-first summer meeting and tenth colloquium of the American Mathematical Society were held at Cornell University, from September 8 to 12, 1925, in connection with the meeting of the Mathe#matical Association of America. The attendance included one hundred and forty-nine members of the society, a record for a summer meeting.

The attending mathematicians and their guests were E very hospitably entertained at Sage College, on the 'beautiful university campus. Many enjoyable social events were arranged by the mathematics department of the university, including a reception at which © President Farrand welcomed the visitors. The joint

dinner of the two mathematical organizations, with Professor H. E. Slaught as toastmaster, was attended by one hundred and eighty-five persons. A hearty vote of thanks was passed to the local members of the committee on arrangements, Professors Tanner, Gillespie and Hurwitz.

The secretary reported the appointment of the following committee on nominations of officers and members of the council and board of trustees for 1926: W. B. Ford, Robert Henderson, D. N. Lehmer, E. J. Townsend and Oswald Veblen (chairman). Professor Harris Hancock was appointed to represent the society at the celebration of the semi-centennial of Vanderbilt University. The council adopted a resolution of thanks to the assistant secretary, Professor Arnold Dresden, for his able and devoted service in carrying on the additional duties of the secretary during the six months' absence of the latter in Europe.

It was announced that the next volume of the Bulletin of the American Mathematical Society will be printed by the George Banta Publishing Company, at Menasha, Wisconsin.

Invitations from Hunter College for the next annual meeting, from Ohio State University for the summer meeting of 1926 and from the University of Wisconsin for the summer meeting and colloquium in 1927 were accepted, with hearty thanks.

The colloquium speakers were Professors L. P. Eisenhart, of Princeton University, and Dunham Jackson, of the University of Minnesota. Each speaker delivered five lectures, which will be published by the society. The subjects were as follows:

Professor Eisenhart: The new differential geometry. (I) Riemannian geometry; (II) Linear connection of a space; (III) Geometry of paths; (IV) Geometry of a sub-space of a linearly connected space.

Professor Jackson: The theory of approximation. (I) The approximate representation of continuous functions; (II) Discontinuous functions and functions of limited variation; (III) The principle of least squares and its generalizations; (IV) Interpolation; (V) The geometry of function space.

The following papers were read at the regular sessions of the society:

Space involutions having a web of invariant rational surfaces: F. R. SHARPE.

Note on six points in a plane and the six conics determined by them: W. B. CARVER.

On the reality of singularities of plane curves: T. R. HOLLCROFT.

Self-projective plane 5-points: LOUIS WEISNER.

Plane cubic curves in the Galois fields of order 2n: A. D. CAMPBELL.

Generalization of certain theorems of Bohl. Second paper: F. H. MURRAY.

Ricci notation for geometrical products: C. L. E. MOORE.

Projection of a fixed vector on a surface: G. Y. RAINICH.

Mass in curved space-time: G. Y. RAINICH. Mutually consistent regression surfaces for three-dimensional frequency solids: B. H. CAMP.

New properties of an orthocentric system of triangles: A. A. BENNETT.

Solutions of the Einstein equations for empty space: H. W. BRINKMANN.

Einsteinian 4-spaces imbedded in euclidean 5-space: H. W. BRINKMANN.

Riemann spaces of class one: H. W. BRINKMANN. The torsion of a Riemannian n-space imbedded in a euclidean m-space (m≤n+2): H. W. BRINKMANN.

Solution of the problem of the thick rectangular plate with two opposite edges supported and two edges free, and under uniform or central load: C. A. GARABEDIAN. Rectangular plates of constant or variable thickness: C. A. GARABEDIAN.

A generalization of the tetrahedral complex in odd Sn-1. Preliminary report: J. A. EIESLAND.

The loci of point singularities on a generalized Kummer surface in odd Sn : J. A. EIESLAND.

- 1

Note on rational plane cubics: C. A. NELSON.

A projective theory of affinely connected manifolds: T. Y. THOMAS.

A study of the conformal mapping w = az + b/z+c/22 and its application to aerodynamics: F. D. MURNAGHAN. On the use of fractions in the algebra of logic: A. D. CAMPBELL.

Concerning the relation between separability and the proposition that every uncountable point set has a limit point: R. L. MOORE.

Concerning the separation of point sets by curves: R. L. MOORE.

The double elliptic case of the Lie-Riemann-Helmholtz problem of the foundations of geometry: R. G. LUBBEN.

Concerning limiting sets: R. G. LUBBEN.

Surrounding theorems with applications to questions of accessibility: R. G. LUBBEN.

Taylor's theorem in general analysis: L. M. GRAVES. On the oscillation of a continuum: W. A. WILSON. Some properties of a continuum limited and irreducible between two points: W. A. WILSON.

A proof of Weierstrass's theorem, with applications to Dirichlet's principle: E. R. HEDRICK and M. B. PORTER.

On a generalization of Gibbs' phenomenon: E. R. HEDRICK and M. B. PORTER.

A problem in the calculus of variations with an infinite number of auxiliary conditions: R. G. D. RICHARDSON. On convergence factors in multiple series: C. N. MOORE.

Relations between the singular points of n ordinary differential equations of the first order: MARSTON MORSE.

A class of reciprocal functions: EINAR HILLE. Boundary problems and expansion theorems in the theory of integro-differential equations: JACQUES TAMARKIN.

Second law of the mean in the theory of definite integrals: JACQUES TAMARKIN and C. E. WILDER.

On a general formula in the theory of Tchebycheff's polynomials and its applications: J. A. SHOHAT.

Note on a fundamental theorem concerning the limit of a sum: H. J. ETTLINGER.

On the conditions of integrability of covariant differential equations: J. A. SCHOUTEN.

The fundamental region for a Fuchsian group: L. R. FORD.

On the form of the solid of revolution of minimum resistance when the normal resistance varies as the nth power (n>0) of the normal velocity: R. P. AGNEW.

Some properties of bounded polynomials in several variables: 0. D. KELLOGG.

Simplification of a general method of summability of divergent series: L. L. SMAIL.

Multiply transitive substitution groups: G. A. MILLER A program on ordinary differential parameters. Preliminary report: O. E. GLENN.

Groups in which the normalizer of every element is abelian: LOUIS WEISNER.

On the formal modular invariants of binary forms: W. L. G. WILLIAMS.

Application of the theory of relative cyclic fields to both cases of Fermat's last theorem: H. S. VANDIVER. On algorisms for the solution of the quadratic congruence: H. S. VANDIVER.

Laws of reciprocity and the first case of Fermat's last theorem: H. S. VANDIVER.

A new theory of the representation of integers as definite quadratic forms: H. S. VANDIVER.

Note on the condition that a cubic equation have an integral root: H. S. VANDIVER.

Definite linear dependence: L. L. DINES.

On certain symmetric sums of determinants: L. L. DINES.

Proof that large primes have four consecutive quadratio residues: A. A. BENNETT.

The algebraic structure of the formulas in plane trigonometry. Second paper: T. H. GRONWALL. An algebra of sequences of functions: E. T. BELL.

The next meeting of the society will be in New York City on October 31; the San Francisco section will also meet at Berkeley on the same date. The western Christmas meeting will be held at Kansas City, in conjunction with the meetings of the Southwestern Section and with the American Association for the Advancement of Science. The annual meeting will be held at Hunter College, New York City, on January 1 and 2, 1926.

MECHANICAL POWER1

I HAVE selected for my address to-night the subject of power-mechanical power-because I believe even workers in science do not fully appreciate the extent to which our present-day civilization is dependent upon this product of science. It is only within a century, however, that mechanical power has become so great a factor in our daily lives.

A century is a very short period in comparison with the number of years man has inhabited this earth. Up to within about a century ago, man truly obeyed the biblical injunction to earn his bread by the sweat of his brow, for the great majority of men and women were slaves or serfs. The Greek and Roman civilizations rested on slavery. Athens had 400,000 slaves to 100,000 free citizens. The industries of Rome were run almost entirely by slave labor.

In the latter days of the Roman Empire, water power became sufficiently developed to compete with slave labor, and "water mills" gradually displaced slave labor in the bakeries, in irrigation and in sawing marble. During the middle ages, mechanical power from water wheels and wind mills was applied in grinding grain, in metallurgical processes and in mining and quarrying, but to a limited extent only.

By the end of the seventeenth century, the coalmining industry reached appreciable proportions in England and on the continent. As the mines were worked to greater depths, the pumping of water from them became a serious problem. The pumps were operated by horses-as many as 500 horses being employed at one mine for this purpose. The expense of pumping became so great that many mines were abandoned. This situation was relieved by the invention of the steam-pumping engine-that of Savery in 1698 and of Newcomen in 1705.

Economic conditions at this time-the first half of the eighteenth century-are indicated by the average wages of a skilled workman in England, about $2.40 a week. Wheat varied from $1.00 to $1.50 a bushel. Thus, the carpenter or mason could earn only from two to three bushels of wheat for his week's work.

Before the eighteenth century, man used only a few elements of machines and crude combinations of them. In the latter part of that century occurred those great inventions in spinning and in weaving where the skill and intelligence of the workman were transferred to

1 Address of the retiring president of the Nebraska Chapter of Sigma Xi on May 15.

machinery operated by mechanical power. The nineteenth century witnessed the development of machinery with greater than human skill and with but little less than human intelligence, many times multiplying man's productiveness through the utilization of mechanical power.

Mill machinery was first operated by water power while the steam engine was first developed to pump water from coal mines. It is therefore not surprising that the first proposal to operate a mill by steam power comprised a steam boiler and engine to convert the heat energy of coal into mechanical energy, a pump to convert the mechanical energy into potential energy of water and a water wheel to convert the potential energy of the water back into the mechanical energy for driving the machinery. The crank connecting rod mechanism was soon invented, however, affording a means of converting reciprocating into rotary motion and thus utilizing more directly the mechanical power of the reciprocating steam engine.

In recent years, we have gone back in many instances to the original proposal, using, however, in place of the water pump, piping and water wheel, an electric generator, transmission line and electric motor between the source of mechanical power and the machinery where the mechanical power is to be utilized. I mention this simply to emphasize the fact that while electricity is often a very convenient means of transmitting power, it is not a primary source of power and it must in general be converted into mechanical power or into heat before it can be utilized.

In 1869, the first year that power statistics were collected by the Bureau of the Census, the mechanical power for the industries of the United States was obtained only from water wheels and from steam engines and boilers fired with coal. The internal combustion engine had not yet reached a practical form. The installed primary power in the manufacturing industries was 1,130,431 horsepower in water wheels and 1,215,711 horsepower in steam engines. In mining and quarrying, the installed primary power was 2,247 horsepower in water wheels and 109,111 horsepower in steam engines. Steam power was used exclusively on the railroads and on ships, amounting to about 3,300,000 and 1,070,000 horsepower, respectively.

In 1869, there was thus available in the United States about 6,827,000 horsepower, of which about 16 per cent. was water power and the remainder steam power. The population was about 38,116,000 people, somewhat less than 0.2 of a horsepower being therefore available for each person.

In the fifty years from 1869 to 1919, remarkable

increases occurred in the horsepower available in prime movers. There have been further advances since that date, but the following approximate figures, based partly on the census data for 1919, will give some idea of the tremendous amount of mechanical power now at the service of mankind. In manufacturing, the 2.3 million horsepower of 1869 has grown to more than 29.5 million. In mining and quarrying, the 111 thousand horsepower has increased to over 6.8 million. On the railroads, the horsepower has increased from 3.3 million to sixty-five million or more. The estimated horsepower of the United States Navy is about ten million, and commercial shipping and private yachts and motor boats will account for another ten million horsepower. On the farms, animal power only was used in 1869, amounting to less than ten million horsepower; in 1919, while the animal power had more than doubled, mechanical power to the extent of about 20 million horsepower was employed. Probably four million horsepower are installed in isolated plants in non-industrial establishments. In 1869, central power stations did not exist for furnishing electricity for lighting, railways, etc.; in 1919 they had an installed capacity of twenty million horsepower, of which about ten million, used in industrial plants and on farms, is included in preceding figures. Conservatively, 345 million horsepower in internal combustion engines are installed in the seventeen million automobiles, motor trucks and tractors in use in this country, of which about six million horsepower is already accounted for as employed in agriculture.

The grand total of these figures is over five hundred million horsepower available for a population of 105 million people, or about five horsepower for each man, woman and child. Since a man's power is less than one tenth of a horsepower, this is equivalent to more than fifty slaves for each inhabitant of the United States.

The ancient Greek triremes had ten marines, twenty sailors and 170 rowers. Compare this with the airplane carrier Saratoga, launched a few days ago, having a crew of 179 officers and 1,695 men and propelling machinery of 180,000 horsepower. This power is equivalent to that of two million galley slaves.

Of the five hundred million horsepower available in 1919, about eight million, or less than 2 per cent., was water power. The remainder had for its primary source of energy, coal, petroleum or natural gas, except about one tenth of one per cent. in windmills on farms. The United States Geological Survey has estimated that the amounts of energy contributed by the four main sources of energy in the United States

rther

of

I

SAY

in 1919 were in the following proportions: Coal, 77.3 per cent.; petroleum, 13.6 per cent.; natural A gas, 4.3 per cent.; water power, 4.8 per cent.

1919, The first three sources are limited and will some day be exhausted. It is often assumed that water power will then be developed to take their places. The potential water power resources of the United States, even with water storage, are estimated by the United States Geological Survey to be only 34,818,000 horsepower, or less than one fourteenth of our present installed capacity. Evidently, water power can never take over the burden now borne by coal, oil and gas. Unless science and engineering develop wider applications of what are now minor sources of mechanical power, the human race must some day return to work.

There is a tendency for the man on the street to shrug his shoulders and say that the scientists will discover other sources of energy before that timethat the energy of the atom will be unlocked, that electricity will be taken from the air. To a person not well grounded in the physics of energy and matter, such propositions do not appear any more wonderful than, for example, the radio. But the fact that Mr. E. W. Rice, Jr., of the General Electric Company, estimates a revenue of several million dollars annually for the sale of electric power to operate radio equipment, is an indication that the laws of thermodynamics are not contravened by this device. Although I am inclined to question whether the second law of thermodynamics is of as broad application as generally stated, I regard as very improbable the unlocking of stores of energy from sources other than those already used to some extent.

Some research should be devoted to the development of what are now minor sources of mechanical power. However, the exhaustion of our natural resources of coal, oil and gas is not immediate. Petroleum in quantity will probably be available for two or three hundred years and coal for two or three thousand years. Greater efforts should be exerted to improve the combustion of fuels, the production of mechanical power from the heat of combustion and the utilization of mechanical power by machinery of various kinds.

Although coal will be available for many years, the better grades for metallurgical and power purposes are approaching exhaustion. Methods of utilizing the poorer grades must be developed to higher degrees of efficiency. One method of accomplishing this is through reducing the amount of inert nitrogen in the air supplied for combustion. There is thus needed an economical process for producing oxygen which may be mixed with atmospheric air or used in a nearly pure state.

After combustion has taken place, the energy exists as heat, and improved means should be sought for converting heat energy into mechanical energy. It has been customary to analyze power plant performance by the first law of thermodynamics, but such an analysis should preferably be based upon the second law in order to show where the greatest inefficiencies exist in the process of converting heat into mechanical work. The first law analysis heretofore made is misleading in this respect. For example, according to the first law, no loss whatever results in a steam power plant by heat transfer from the products of combustion in the furnace to the water in the boiler to evaporate it into steam. The second law analysis shows this to be the largest single item of loss in the whole power plant. The subject of heat transfer is thus an important matter of research for improving the production of mechanical power from heat.

Research in utilizing mechanical power is required in order to minimize the expenditure of fuel for the accomplishment of a desired result. While there are a multiplicity of ways of using power, the main item of such research should be friction, for nearly all the energy of mechanical power is finally dissipated as heat in overcoming frictional resistances. This may appear to be a startling statement; yet if you will think of the movement of a railroad train from Lincoln to Chicago through the burning of say thirty tons of coal under the locomotive boiler, you will appreciate that the tractive effort of the locomotive is expended entirely in overcoming the frictional resistances encountered by the train against the air and rails and in the bearings. In industrial processes in general, a negligible amount of mechanical power is stored up in some form of available energy; nearly all of it is dissipated in overcoming friction. Research upon the frictional resistance of fluids flowing through pipes, of bodies moving through fluids and of the lubrication of bearings is thus of the greatest importance in prolonging the life of our fuel re

sources.

Economic conditions are bringing about improvements in the production and utilization of mechanical power. For example, the coal consumption per kilowatt hour produced by electric utilities dropped 25 per cent. in the five years from 1918 to 1923. Also, steam locomotives have been developed which, in comparison with the standard locomotives of but three years ago, will haul 50 per cent. more tonnage for the same amount of coal burned. Until to-day, coal economy has been of very minor importance in railroad operation; but conditions are changing, and scientific analysis is now being applied to the steam locomotive with very large improvements in efficiency.