ances were effected by buying annuities for one year 

 certain, to commence at the death of a person named. 

 This had the effect of putting an end to such annuities. 



C 2. The form of the rule for computing the value of 

 fines, and its introduction into the method of calculating 

 the present value of a perpetual advowson (pp.231. 236. 

 and Appendix the Second). It will be found that the 

 rule of every writer on the subject is palpably wrong 

 in principle, with the exception of that of Mr. Milne. 



3. The rule for the valuation of uniformly increasing 

 or decreasing annuities, given in the fifth appendix. A 

 simple application of the differential calculus is made a 

 striking instance of the position, that the labour of a 

 person of competent knowledge is seldom lost. The 

 annuities given by Mr. Morgan and Mr. Milne, are for 

 every rate of interest, from three to eight per cent.; 

 and perhaps those gentlemen may have had some doubts 

 as to the necessity of inserting the two last rates. It 

 now appears, however, that, in consequence of the extent 

 to which their tables are carried, the values of increasing 

 or decreasing annuities, can be calculated with great 

 accuracy for three and four per cent., and with sufficient 

 nearness for five per cent. ; and with very little trouble, 

 compared with that which it must have cost Mr. Morgan 

 to calculate the table referred to in page xxviii. of the 

 Appendix. 



The rules, in page xxix. of the Appendix, contain a 

 point which, as no demonstration is given, may cause 

 some difficulty. In turning an annuity or insurance 

 which cannot be extinguished during the life of the 

 party into one which can, a direction to add is given 

 which will at first sight, perhaps, be supposed to be a 

 mistake, and that subtract should be written instead. But 



