The results of massing one ingredient upon another may be illustrated by Diagram A, which is familiar to all students of economics. With a given quantity of alcohol let us mix varying quantities of acid, which we shall represent on the line OX. The quantity of the product, ether, we shall represent on the line OY. When a quantity of acid represented by the line OC is put into the mixture, let us assume that we get a quantity of ether represented by the rectangle OABC. Twice that quantity of acid with the same quantity of alcohol will increase the product, ether, but will not double it; that is, the product increases but not in proportion to

Y

A

B

D

E

G

H

F

Diagram A

-X

the acid. Let us suppose that a quantity of acid represented by the line OF produces, with the other ingredients, a quantity of ether represented by the rectangle ODEF. A third increment and a fourth would still result in some additions to the product, as long, perhaps, as any of the original quantity of alcohol was able to escape the mass action of the acid. Eventually the point would be reached where further increases of the acid would add nothing to the product.

It will be observed, however, that the addition of the increment CF to the acid did not add the rectangle CIEF to the product. The addition to the product is the difference between the rectangle OABC and the rectangle ODEF. That difference is represented by the rectangle CGHF.

The marginal product. This is technically known as the marginal product of the acid. This technical term does not mean, however, that even the product CGHF was produced by the acid alone; it merely means that whatever value there is in the added product CGHF would be the outside limit of the value of the added ingredient CF.

Air and gasoline in a carburetor. A problem something like this presents itself in practical form in the use of air and gasoline in an internal-combustion engine. Both are necessary, but they may be mixed in somewhat variable proportions. One may use a rich or a lean mixture. A rich mixture is one rich in gasoline and lean in air. A lean mixture is one lean in gasoline and rich in air. Combustion itself is a chemical process and presumably follows the law of definite proportions rather than the law of variable proportions. But the mixture of air

are actually combined. A lean mixture masses air on the gasoline and enables more of the latter to burn, though much of the air is unburned; a rich mixture does not mass so much air, does not burn so much of the gasoline, but burns a larger proportion of the air. If air were expensive and gasoline cheap a rich mixture would be more economical. Since air costs nothing and gasoline is expensive a lean mixture is the more economical. The leaner the mixture that can be made to explode, the greater the economy of gasoline. It wastes air, but that is not bad economy. In short, we try to adjust our carburetors so as to approximate as nearly as possible to the conditions represented in Diagram B.

Let us assume that a quantity of acid represented by the line OL results, under certain conditions of manufacture, in a quantity of ether represented by the rectangle OJKL, while a

quantity represented by the line OQ results, under similar circumstances, in a quantity represented by the rectangle OMNQ. But these two rectangles are equal; that is to say, with a quantity of acid equal to OL one gets precisely the same as with OQ. In short, the additional acid, LQ, is thrown away. It is of no use whatever in that particular mixture, and yet, the acid being all of uniform quality, it is as good as any of the rest. The average product, however, for that quantity of the variable ingredient would be represented by the rectangle LPNQ. It would be foolish to pay that much for it, however, or, if it cost as much as that quantity of ether would sell for, it would be foolish to use so much. If, however, it cost absolutely nothing it might pay to use that much or nearly as much in order to be sure of getting the full use of the alcohol, which is expensive.

If we were to reduce the broken lines which form the tops of the rectangles in the two diagrams A and B to smooth curves, we should get something like the following:

ent) gradually falls; but as long as there is any product whatsoever there must be an average productivity per unit of that ingredient, this average productivity being represented by the descending curve YB. But the marginal productivity falls much more rapidly and may even become a minus quantity. When so much of this variable ingredient is used as to yield the maximum total product, and further additions add nothing to the total, then these further additions are said to have a marginal productivity which is nil. In Diagram C the marginal

product of varying quantities is represented by the line YA. In some mixtures further additions may actually interfere with the work and reduce the total product. The curve YAC represents the marginal product under these conditions. In other mixtures the excess of the variable ingredient does not become positively detrimental or destructive, but merely neutral. In such cases its marginal productivity becomes nil but never a minus quantity. The curve YAC in Diagram C, in order to represent this class of cases, would have to be redrawn. It should never fall below the line OX.

Reversing the experiment gives corresponding results. If now we change the experiment and introduce varying quantities of the other ingredient in the mixture with a fixed quantity of the ingredient which we have been considering as the variable factor, we shall get results which harmonize perfectly with those which we have secured hitherto. Returning to the case of alcohol and acid in the making of ether, let us start with a quantity of acid represented by the line OL in Diagram B. According to our assumption, as explained earlier, that quantity of acid with the original quantity of alcohol produced no more ether than did a slightly smaller quantity of acid represented by the line OL. If now we mix a quantity of acid equal to OL with enough additional alcohol to bring the mixture to the same proportions as in the original mixture, in which OL acid was used, the product, ether, will increase in exact proportion to the increase in the alcohol, provided, of course, the reaction is not hindered by the smallness of the receptacle or by some other extraneous circumstance.

To use, for example, a fixed quantity of air for each explosion, but a larger quantity of gasoline, would require a larger cylinder. Making such necessary allowances, we can say that if the maximum amount of air in a gasoline engine is used with a given quantity of gasoline, so that more air would be of no advantage whatever, then a little more gasoline could be introduced and would add considerably to the power. There being enough air in the mixture to get the maximum combustion

of gasoline, the power would for a time increase in proportion to the gasoline. As more and more gasoline is introduced, however, with a fixed quantity of air, making the mixture gradually richer, a smaller and smaller proportion of gasoline will be burned because of a scarcity of air. If the mixture is made rich enough a point will be reached where further additions of gasoline will add nothing whatever to the power. The marginal productivity of gasoline is then nil. When the mixture gets so rich that it will not explode, it reduces the power, and the marginal productivity of gasoline becomes a negative quantity.

The marginal product of each factor the complement of that of the other. The marginal productivity of each factor in the combination is, it will be observed, the complement of that of the other factor. When the proportions are such that the marginal productivity of one is nil, that of the other is 100 per cent of the average product; that is, the total product increases in exact proportion as this factor is increased. When the proportions are such that the marginal product of one factor is low, that of the other is high, the sum of the two marginal products always equaling the total product.

When there are more than two factors in the compound the problem becomes more complicated, but the principle is the same. In such a case it is better to treat each one separately, regarding all the others as a bunch, or cluster, and thus treating them as a unit. Marshall has suggested the word "dose" to designate a group of factors. Thus, if we were considering nitrogen, phosphorus, potassium, and all other factors in soil fertility, we could treat all the factors except, say, nitrogen as constants. By varying the nitrogen in the compound we get variations in the crop yields.

Rothamsted experiments. Experiments of this kind have actually been carried on at the Rothamsted estate, near London, where the great work inaugurated by Sir John Lawes has been carried on for many years. In one experiment, for example, five plots of land of approximately equal fertility were treated alike in all particulars save one. Different quantities