Suppose that the haul on the home road is 200 miles and that on the foreign road is 50 miles. In this case the home road will receive 200/(200+50) =80% of the through rate. Suppose that the home road is lengthened five miles, then it will receive 205/(205+50)=80.392% of the through rate. Assuming as before that the total payment on a given shipment of freight is $1, the home road will receive by the first plan 80 c, or at the rate of .4 c. per mile. Adding five miles, the home road will receive only .392 c. in addition, which will be at the rate of .0784 c. per mile for the additional five miles. This is only 19.6% of the original rate, and is but little over one-third of what the additional distance will actually cost. In such a case, although there is some compensation for the additional distance, it is not sufficient to pay for the added cost. A study of the two numerical problems given above will show that there is some proportion of home haul to foreign haul at which the added receipts will just equal the added cost. When the home haul is a very large proportion of the total haul there is a loss, although not a total loss. When the haul is entirely on the home road, which is the case of competitive local traffic, the added distance is a complete loss without any compensation.
When the added distance is a large part of the home haul and the foreign haul is very large, then the profit of the home road is considerable and the total transaction is distinctly profitable. A further development of this course of reasoning might be an interesting mathematical study, but its precise application is useless for the reason given below.
Every station on the home road has at least potential traffic relations with every other station on every other road in the country. The traffic between each station and every other station presents a new combination. The effect of an increase of distance on one branch of the traffic will be more or less compensated by an opposite effect on the traffic between two other stations. The net effect on the total receipts of the road could only be obtained by the solution of a problem with a very large number of elements which are not even constant, but which are subject to unforeseen changes. Therefore the only practical use that we can make of a demonstration like the above is to derive from them certain general conclusions as follows.