308 Mr. Adams on the perpetual Renewal of Leases. [Oct. 



allowed to replace it with the best they can find ; to determine 

 the probable time between each nomination. 



Solution. — The present value of 1/. payable every x years for 



ever at e per cent, compound interest is equal to - ; — - 



(Baily on Interest and Annuities, p. 77.) The present value of 

 all the renewals for ever is equal to 3 (— : — l\ where A de- 

 notes the value of an annuity of the lives proposed to be added. 

 — (Baily on Life Annuities and Assurances, p. 174.) 



= 3 



(1 + ey- 1 



Hence we have the following equation, viz. 

 (— — l) = m, from whence (1 + e) x = 1 + -, and per loga- 

 rithms x x I (1 + e) = I . (] -\ — J .*. x sa \ + m) . 



a gene- 



l (1 + «) 



ral theorem for the times between each nomination. 



Since A, according to the tables given in vol. ii. Baily's 

 Annuities, may have five different values, x, in consequence, will 

 vary accordingly, which values are selected and placed in the 

 following table : 



Each of these means corresponds nearest to Table XXVII. 

 Hence the probable time between each nomination, according to 

 the above table, is about 15 years. 



By the Carlisle Tables, as given in Mr. Milne's Annuities, the 

 value of A at 4 per cent, is 19*792 years, and at 5 per cent. 

 16-79, the corresponding age being seven years ; from whence 

 the value of x, according to the former, is 20*8656 years ; and, 

 according to the latter, 20*6852. 



