We have in general ,
and in this particular case
Whenever, therefore, the ratio only differs from unity by a quantity less than the cost of transportation of a monetary unit from one place to the other, the balance between the two places will be settled without actual transportation of money and by the mere effect of the rate of exchange.
Let us now suppose any number of places in correspondence, and let mi,k express generally the total of the sums annually due from the place (i) to the place (k), and ci,k the coefficient of exchange from (i) to (k). The number of these coefficients for a number of places (r) will be r(r—1); but, as in general, , the number of coefficients to be determined is reduced at the outset to .
Nor are all these coefficients mutually independent; for if, for instance,
any one having to convey money from (k) to (i), instead of getting draft from (k) on (i), will find it cheaper to get one from (k) on (l) which he can exchange for one from (l) on (i). For the same reason it is impossible to have