APPENDIX TO CHAPTER XIV 397 case, it is the capital-value of such an income stream which will vary from time to time. As has been seen, the capitalvalue of such an income stream is found by dividing the rate of income a by the average of the individual rates of interest, such as, in the above example, 10%, 5%, 6 %, etc., ad inf., the average being obtained by the formula given in Appendix to Chap. XII, § 5. If such average rate of interest be called jj, the . a . capital-value will be =. Suppose, for instance, that the person J has a uniform perpetual income of $5 per annum. If the rate of interest to-day, j,, is 5%, the capital-value to-day will be $100. If next year, j, (the average of the future rates in individual years, beginning at that time) is 4.99, the capital-value will be $102. If, two years from date, j, be 5.19, the capital-value will sink to $98. Adopting such an income stream as a standard, the propositions as to impairment or increase will still be true, provided such impairment or increase is measured with reference to the variable capital-value just shown. Thus, if at the end of the first year more income than $5 is received, the capital-value will be impaired by the difference, this impairment to be reckoned with respect, not to $100, but to $102, which would be the value had the income remained standard. Thus, the effect of a difference between real and standard income may be stated in the same terms, whichever of the two definitions of standard income is adopted. In the one case the standard is with reference to constant capital and variable income; in the other, to variable capital and constant income. In practical life, the former standard is usually employed, although for certain purposes the latter would be more suitable. We all know of cases of investors who, twenty years ago, invested at a high rate of interest, and who have taken pains merely to maintain the value of their capital unimpaired, although they were well aware that the rate of interest was constantly sinking. In consequence, these persons are now forced, when reinvesting, to suffer a large decrease in income, which could have been avoided had they kept in view the maintenance, not of their capital, but of their income, and laid aside each year a certain sum in order to offset the fall in the rate of interest. The reason such a procedure is not common is