§ 7 (to Chapter II, § 5) P must be a Specific Form of Average in order to vary directly as M and V and inversely as the

Let us assume that V and the Q's remain invariable while M changes to Mo and p, p', p', etc. to po, p'o, p'o, etc. (The subscripts "0" refer to a year called the base year other than the original year.) We have for the two years respectively the two equations:—

whence by division, we obtain

The last expression is evidently a weighted arithmetical average of

etc., the weights being poQ, p'oQ', etc. We conclude that, if the velocity of circulation and the quantities of goods exchanged remain unaltered, while the quantity of money is altered in a given ratio, then prices will change in this same ratio "on the average," the average being exactly defined as a weighted arithmetical average, in which the weights are the values of goods sold, reckoned at the prices of the base year. The ratio may evidently also be written:—

which is a weighted harmonic average of

etc., in which the weights are pQ, p'Q', etc., that is, the values, not in the base year, but the other year.

If M and the Q's remain invariable, while V changes from V to V1, evidently the ratio V/V1 will be expressed by precisely the same formulæ as above.

If the Q's remain invariable, while M and V both change, evidently the ratio

will be expressed by the same formulæ.

Again the same formulæ apply if M and V remain invariable while the Q's all vary in a given ratio, or if the Q's all vary in a given ratio in combination with any variation in M or V or both. In short, the formulae apply perfectly in all cases of variation, except when the Q's vary relatively to each other.