ON THE VALUE OF REVERSIONS. 



231 



will be useful in the calculation of the values of advow 

 sons, with the correction above proposed. 



QUESTION. At the end of the year in which A dies, 

 a fine of 11. is to be paid, and a new life nominated, of 

 which the value will then be P : at the end of the year 

 in which P dies, another fine of II. is to be paid, and a 

 new life P nominated, and so on for ever : what is the 

 present value of all the fines, or what present money 

 must a person be considered as paying who receives an 

 estate charged with the preceding liabilities ? 



This problem, as will be more fully explained in the 

 second appendix, was incorrectly solved by every writer 

 on the subject, down to the time of Mr. Milne, whose 

 solution, though perfectly correct, is in a difficult form. 

 The coincidence of the rule I now give with that of 

 Mr. Milne will be shown in the appendix cited. 



Let us suppose a fine of IL per annum, first payable 

 at the end of the year in which A dies. If, then, a 

 receiver P were appointed for his life, his interest in the 

 fines, at the end of the year in which A dies, would be 

 1 + | P ; and if at his death a second receiver were ap- 

 pointed, of the same age at which the first was when his 

 term began, the interest of this second receiver at his 

 entrance would also be 1 -f |P, and so on. But if the 

 tenant compounded with each receiver on his entrance, 

 for the; rents payable during the life of that receiver, it 

 would evidently be equivalent to paying a fine of 1 -j- |P 

 at the end of the year in which each dies, and also at 

 the end of the year in which A dies. But the present 

 value of all the rents is a perpetuity diminished by the 

 value of an annuity on A's life, or | |A, And if this 

 be the value of a fine of 1 + |P, then | | A, divided by 

 Q 4 



