under state laws and have no additional municipal regulations. A number of cities such as Baltimore, Buffalo, Los Angeles and Milwaukee have local provisions in addition to state laws. In 1910, sixteen states had laws providing for the state inspection of meters and of the purity of gas-Connecticut, Georgia, Kansas, Maryland, Massachusetts, Nevada, New Hampshire, New Jersey, New York, Ohio, Oklahoma, Vermont, Virginia, Washington, Wisconsin and California (B. of S. Circular No. 32). Doubtless the list is now larger.

In 1910 the net income to the state of Massachusetts in meter-testing alone was over $5,000. The total cost of the tests on quality, purity, pressure, etc., was assessed on the operating companies according to their sales. Meter-testing is on the fee basis. There is no good reason why such a department in North Carolina would not yield a revenue to the state.

That the Corporation Commission in North Carolina should have the power and machinery at its command to protect the interests of citizens seems obvious for the following reasons: Under existing law it is the duty of the commission to regulate the rates to be charged by gas companies. The proper price is determined in a large measure by the quality of product sold and this is almost at the will of the producer. Gas in New York City furnishes 680 heat units per cubic foot and is sold at 80 cents. Gas in Durham furnishes at times less than 500 heat units and is sold at $1.50. In one city in this state gas furnishing 412 heat units sold for $1.60. The standard requirement in regulated states is around 600 heat units. The difference in quality means a loss of from five to twenty thousand dollars per year to consumers in various towns of this state and the loss would easily run into hundreds of thousands to the state at large. While it may be to the interest of certain communities to sell a cheap, poor gas it is safe to say that it is always against public interest to have a cheap, poor gas sold at a rich, high price. To fairly meet its responsibility the commission must know from its own tests the quality of the product sold. The consumer is entirely helpless.

Aside from the question of rates, the public is vitally interested from the standpoint of health. In the method of manufacture used by one company in this state, carbon monoxide and hydrogen are produced in equal quantities. Both of these gases are odorless and one is a deadly poison. Combined they give a cheap gas furnishing about 300 heat units. This gas causes a meter to register just as fast as a 600 heat unit gas. It is the duty

of this company to carburet this gas with an oil which not only brings its heat value to standard, but gives it a very pungent odor that makes it noticeable in case of a leak. In this town a series of fatal accidents have occurred due solely to the neglect of the service company. In other methods other deleterious elements are introduced by carelessness so that in all cases public interest demands systematic testing under the authority of the state.

It is just as reasonable to let manufacturers sell anything called fertilizer without tests as to composition as it is to permit of the sale of untested gas. Our duty to test meters is just as obvious as our duty to test weights and measures.

The advantages resulting from such an act would not even be principally with the citizen. An expert employed by the state to travel from plant to plant observing and testing, corrects irregularities and errors in manufacture that may mean thousands of dollars saved to the companies. If ammonia appears in the gas it means that a valuable by-product is being lost. So it is with other errors of manufacture. The fact that meters are tested by the state brings a feeling of confidence to the consumer that is worth much to the gas companies. Uniform, improved and economical manufacture brings new and profitable business and this more than compensates for any costs involved.

No abstracts have been received for the following papers:

The relative toxicity of uranium nitrate in animals of different ages, by Wm. DeB. MacNider. Trembles, by Frederick A. Wolf.

Permanency in fleshy fungi, by H. C. Beardslee. Sound-wave photography (lantern), by Andrew H. Patterson.

Evolution in sponges and changes in classification, by H. V. Wilson.

The revision of the atomic weight of zirconium, by F. P. Venable and J. M. Bell.

Recent investigations about cottonseed meal, by W. A. Withers and F. E. Carruth.

The physics of the shrapnel shell, by Andrew H. Patterson.

Portolan charts (lantern), by Collier Cobb. The idea of force in mechanics, by Andrew H. Patterson.

The times we think in, by George W. Lay. The life history of the pecan trunk borer, by R. W. Leiby.

E. W. GUDGER, Secretary

FRIDAY, AUGUST 31, 1917

CONTENTS

LIEBIG'S LAW OF THE MINIMUM IN RELATION TO GENERAL BIOLOG

ICAL PROBLEMS 1

THE Law of the Minimum has never been accurately defined, although the idea that it involves is relatively simple. Professor B. E. Livingston says in a recent paper2 that "this principle is still quite incomplete logically and its statement will assuredly become much more complex as our science advances." In order to get a clear understanding of the law so that it may be stated accurately, we will begin with a simple application to chemical reactions.

One molecule of KOH reacts with one molecule of HCl to form one molecule of KCl and one of H2O. If only one molecule of KOH is present, only one molecule of KCl can be formed, no matter how many molecules of HCl are present; and likewise if only one molecule of HCl is present, only one molecule of KCl can be formed, no matter how many molecules of KOH are present. By considering the weights of the reacting substances, the situation is somewhat complicated: 56.1 grams of KOH react with 36.5 grams of HCl to form 74.6 grams of KCl and 18 grams of H2O. In round numbers 3 parts by weight of KOH and two of HCl give 4 parts by weight of KCl and one of H2O: 3/4 gr. of KOH and 1/2 gr. of HCl are necessary to form a gram of KCl. Let us call these fractions, 3/4 and 1/2, the specific reactive weights of KOH and HCl in respect to the formation of a unit quantity of KCl. Suppose x amount of KOH and y of HCl are given. If x and 1 Paper read before the Biological Club of Yale University, April 19, 1917.

2

2 Plant World, 20: 1–15, 1917.

y are divided by their respective specific 4 reactive weights, we get x and 2y. The 3 smaller of these quantities is a direct measure of the weight of KCl that can be formed from x KOH and y HCl. If, for example, x and y are both equal to three grams, four grams of KCl can be obtained.

These facts can be generalized. If A, B and C are substances which react to form S and u A, v B and w C are necessary for the formation of a unit amount of S, then u, v and w may be called the specific reactive values of A, B and C, respectively. They may be weights, volumes, numbers of molecules or what not. In any particular case, where pA, qB and rC are reacting, the amount of S formed is the smallest of the fractions p/u, q/v, r/w. When the amounts of the reacting substances are divided by their specific reactive values, the smallest quantity so obtained is equal to the amount of the product formed.

This conclusion is directly applicable to the problem of fertilizers. It is known that most of the higher plants must obtain seven elements in combined form from the soil. They are S, P, N, K, Ca, Mg and Fe. If aS, BP, yN, SK, Ca, ¿Mg and Fe are required for a unit amount of growth in some particular plant, say wheat, and if aS, bP, cN, dK, eCa, ƒMg and gFe are present in a particular soil in available form, the maximum amount of wheat that can be grown in that soil will be the smallest of the fractions a/a, b/ß, c/y, d/d, e/e, f/5, g/n. In this case a, ẞ, y, etc., may be called specific growth values for the plant under consideration. When the available amounts of the essential inorganic food constituents are divided by their respective growth values, the smallest quantity obtained gives the maximum amount of growth possible. It was in this connection that Liebig3 first "Die Chemie in ihre Anwendung auf Agricultur und Physiologie," 7te Auflage, 2: 225, 1862.

formulated the Law of the Minimum which, as commonly stated,' says that "the yield of any crop always depends on that nutritive constituent which is present in minimum amount." The use of the term minimum is not strictly accurate, as can be seen from the example of KOH and HCl. If three grams of each are present, the amount of KOH determines the yield of KCl, although both HCl and KOH are present in equal amount. However, the above statement of the law is convenient because of its simplicity.

A much broader application of the Law of the Minimum was indicated by the work of F. F. Blackman, whose conclusions are summarized in his paper on "Optima and limiting factors." Blackman called attention to the complexity of the process of carbon assimilation, the rate of which depends on at least six factors

1. Temperature,
2. Light intensity,

3. Carbon-dioxide supply,
4. Water supply,
5. Chlorophyll,

6. Enzymes.

Where it is possible to vary one of these factors independently of the rest, its effect on the rate of assimilation can be measured, under suitable conditions, and a curve plotted. In this way a temperature-assimilation curve, a light-assimilation curve and a carbon-dioxide-assimilation curve can be constructed. The other factors are more difficult to control. The following curves were constructed by Blackman and Smith from a study of the rate of assimilation in Elodea.

The light curve and the carbon-dioxide curve are straight lines. The rate of assimilation varies directly with the inten

Cf. F. Czapek, "Biochemie der Pflanzen,'' 2: 841, 1905.

5 Annals of Botany, 19: 281-295, 1905. & Proc. R. Soc., B., 83: 389-412, 1910.

FIG. 1. Effect of external factors on assimilation in Elodea. (After Blackman and Smith.) law of reactions for temperatures under 30° C. Above this, the rate of assimilation at first rises and then falls off, the process being complicated at high temperatures by

a "time factor." The same effect has been observed at high light intensities, and with strong concentrations of carbon-dioxide which have a narcotic effect.

Disregarding these complications, we will confine our attention to the first parts of these curves. The ordinates of all three curves are the same, namely, rates of carbon assimilation, which can be measured in terms either of CO2 absorbed or of sugar produced. The former happens to be the more convenient measure. At any given temperature, the rate of assimilation which is a function of that particular temperature can be determined directly by the curve and is equal to a certain distance measured off from the origin on the Y-axis. Similar distances are given for any definite supply of carbon dioxide and for any degree of illumination. In any actual environmental complex, where the temperature, light and carbon-dioxide supply are known, the rate

of assimilation is equal to the shortest distance measured on the Y-axis. This is stated as a general principle by Blackman as follows: "When a process is conditioned as to its rapidity by a number of separate factors, the rate of the process is limited by the pace of the 'slowest' factor." The factor which gives the shortest distance on the Y-axis-that is, the "slowest" factor, he calls the limiting factor.

As a matter of fact the carbon assimilation of green plants is usually limited by the seasonal variation in temperature and the diurnal variation in light, the CO2 content of the air being constant. Nothing has been said of the other factors that effect carbon assimilation-the water supply, chlorophyll and enzymes. These so-called "internal" factors, as well as the "external" factors, are governed by the Law of the Minimum. Of the internal factors, water and chlorophyll are present in exassimilatory enzymes being the only probcess in healthy green plants, the amount of

able limiting factor.

It is not necessary to adduce additional examples to show that the Law of the Minimum is a universal law, affecting not merely the concentration of reacting substances, but all factors that in any way influence a reaction or process. The law is applicable to physical, chemical and geological as well as biological problems. An interesting instance of its application to a problem in physics is the determination of the magnitude of a thermionic current. This varies with changes in temperature, and also with changes in the voltage applied. The temperature formula gives one value, the voltage formula may give another; the lesser value determines the current flowing. The 7 A timely application may be made which is worth bearing in mind. The efficiency of a nation at war is subject to the Law of the Minimum. Defeat, in the last analysis, may be attributed to the effect of some limiting factor.

application of the Law of the Minimum has been worked out in many cases and has been of great use in the interpretation of complicated relations; but it has been recognized as a law and has been consciously applied by plant physiologists and physiological chemists only. Without doubt it can be used to advantage in many problems of the physiology, morphology and ecology of both plants and animals.

The Law of the Minimum must be taken into account in all experimental work, for which it serves both as a precaution and a guide. When investigating the effect of an external factor such as temperature, light, etc., on any given process, it is necessary to keep all other variable factors constant, and then to determine the effect of changes in the factor under consideration. What results might be obtained when this method is used in studying carbon assimilation? Suppose the CO2 supply and the light are kept constant, while the temperature is varied. If the CO2 supply is such that it becomes a limiting factor when the temperature rises above 10° C. then the rate of assimilation will rise with the temperature up to this point, but will remain constant at all higher temperatures, until the destructive effect of the high temperature is manifested and the curve again falls off. Above 10° C. variations in the temperature have no apparent effect under these experimental conditions. But if the CO2 supply is increased so as to permit more rapid assimilation, then the temperature curve can be extended. Negative results from such an experimental method are therefore without significance. It is not enough that the experiment be conducted under constant conditions; the constant

2

factors must not interfere in any way with the carrying out of the process; that is, they

8 Cf. the work of L. B. Mendel, T. B. Osborne and their pupils.

9 Cf. B. E. Livingston, loc. cit.

must not be limiting factors. On the other hand, it is a simple matter to determine by the shape of the curve whether any other factor than the one under investigation is a limiting factor. Such is always the case when a break occurs in the curve; usually the curve rises at first and later runs parallel with X-axis. Such curves were obtained by Miss Matthaei10 in studying the carbon assimilation of cherry laurel at varying temperatures with unit light intensity. The problem is much more complicated, however, when variation of one factor is accompanied by changes in one or more other factors. This complication arises in plotting the temperature curve for enzyme activity. The curve rises at first according to van't Hoff's law of reactions, but eventually a maximum value is reached and the curve falls off. At some point near the end of the ascending portion of the curve a

FIG. 2. Effect of temperature on the activity of malt diastase. (After Kjeldahl.)

break occurs: for all temperatures below this point, temperature is the limiting factor and determines the activity of the point, not temperature, but the amount of enzyme; for all temperatures above this enzyme is the limiting factor. The higher temperatures cause a permanent inactivation or decomposition of the enzyme so that its activity is conditioned only secondarily by the temperature. There is also a time

factor involved here; the longer the temperature acts, the more the enzyme is decomposed, within certain limits. The study 10 Phil. Trans., B, 196: 47-105, 1904.