ON ANNUITIES. 209 



to A an annuity on his own life, withdrawing it as long 

 as either B or C survives with A ; that is, withdrawing 



|AB: AC or | AB + | AC- | ABC. Hence, 



B:C|A=|A-|AB-|AC 



PROBLEM. Required C| A : B, the value of an annuity 

 to be paid as long as either A or B shall survive C. 

 Grant one annuity to be paid as long as A or B is alive 

 (| A : B), and withdraw it as long as A or B lives with 

 C; that is, withdraw |AC : BC. Hence, 



C|A: B=|A:B-|AC: BC 



= |A + |B-|AB-(|AC 

 = |A+|B-(|A-B + |AC 



The same case also amounts to granting |A : B : C and 

 withdrawing |C. 



PROBLEM. An annuity C|A : B, payable as long as 

 either A or B shall survive C, is to be divided equally 

 between them, while they both live, and is then to go to 

 the survivor. What is the value of the interest of 

 each ? 



The interest of A is an annuity of half a pound 

 while both A and B survive C, and of a whole pound 

 as long as he shall survive both C and B : or 

 But, 



i(C|AB)HK|AB)-J(|ABC) 



C: B|A=|A-|AB-|AC+|ABC 

 the sum of which is 



|A-i(|AB)-|AC+J(|ABC) 

 and similarly, the value of B's interest is . 

 |B-i(|AB)-|BC+4(|ABC) 



the sum of which makes up, as it should do, the value 

 of C|A : B, as given above. 

 p 



