ON ANNUITIES. 



203 



which is to be paid during the life of C, after the 

 deaths of A and B, if A die before B. 



A : B : Cjl E is the present value of 11. to be paid 



at the end of the year in which the last of the lives, 

 A, B, C, drops,, on condition that B shall have died second 

 or jhircL, and that E shall be alive. 



A|P denotes the present value of an annuity which 

 is to begin payment at the end of the year in which A 

 dies, and to last during the life which shall then have 

 the value P. If there be several conditions, put a symbol 

 over the status which ends and before the one which 

 begins. Thus, 



| ABC : P denotes the value of an annuity which is 

 to be paid during the joint existence of A, B, and C, 

 and further during that of a life which is to be 

 nominated (then having the value P) at the end of 

 the year in which either of the three drops : while 

 ABC|P denotes that part of the annuity which is paid 

 during the life of P. An author's interest in his 

 works, which is now denoted by | A : 28 was proposed, 

 in a bill lately before the House of Commons, to be 

 changed into 'A : 60. 



The preceding notation will admit of almost any 

 degree of extension, and will be found perfectly capable 

 of expressing any case which now occurs in practice. 

 It must receive some generalisation before it can be 

 applied to an indefinite number of lives, in the manner 

 of Mr. Milne. But since it very rarely happens that 

 more than three lives occur in a practical question, I 

 shall leave farther extension to those who may find the 

 want of it. I shall merely now add, that any case 

 must admit of expression by means of a notation which 

 provides for the conditions under which the benefit 

 begins, the number of years which it is to last, and the 

 conditions of discontinuance : and also that in problems 

 of any degree of complexity, the invention of the 

 notation will be a useful preliminary to the actual solu- 



