a few of the more simple ones. Several attempts have been made to treat political economy mathematically, but they have chiefly resulted in failure, for the reason that the mathematics has taken quite a subordinate part, being used to express the result of elaborate reasoning by words. It is like the man who keeps a watchdog and does the barking himself.

The most successful attempt so far seems to have been made by Professor Jevons,^* but he states in his preface that, although many of the problems might have been solved more directly, he preferred to limit himself to the simplest possible mathematics, thus the book hides rather than shows the value of applying mathematics to the subject. Another writer on the subject is Professor J. D. Everett.^† A long list of other writers is given at the end of Professor Jevons's book, but the two mentioned are the only mathematical ones to which I have been able to refer; and, from a remark on the customary method of treatment in Professor Everett's paper, I believe that the proofs in the following paper are new, though the results have in many cases been previously obtained by a patient application of logic.

The fundamental principle which is assumed in the following is that in the serious affairs of life a person always endeavours to obtain the maximum return on an investment. This one might almost call an axiom, and as such it is used. With regard to the definitions, I have defined the quantities as I intend to use them, and as long as a definition and its use are consistent no more is required of it.

Many people think that the application of mathematics to political economy is an almost impossible proceeding. The science, they say, is too vague and conditional for it to be possible. The same might have been said of other sciences in their beginnings, but which have since had mathematics successfully applied to them. For instance, what is more capricious than evolution? yet Professor Pearson is successfully applying mathematics to this subject. The problems of political economy in many cases resemble problems in dynamics, and it is quite a possibility that its elements might be expressed in terms of energy which would thus bring it more into line with other branches of applied mathematics. In fact, so apparent are the advantages of the mathematical treatment of the subject to many that a well-known professor jokingly said, in a lecture on the representation of facts by curves, that before long we should probably see our legislators,