We now proceed to compute the value of annuities on the longeft lives of two lives, on the fame principles as before; but we fhall not give the investigation of the theorems or rules, but only the rules themselves, becaufe of the intricacy and difficulty attending it, this work being not defign'd for Algebraifts; and fecondly, because it would take up more room than we can poffibly fpare for this purpose.

F

CASE I. If both the lives are equal.

RULE. ft. Divide the complement of a fingle life by 6 times the amount of 11. for one year; 2d. fubtract the quotient from the value of one of the lives; 3d. add half the remainder to the value of one life, and the fum is the va lue of both lives.

QUEST. What is the value of the longeft of two equal lives each of 48 years; allowing compound intereft at 4 per cent. per annum?

Amount of 11. for 1 year 1.04

86

་

CASE II. If the lives are both unequal. RULE. ft. Divide the complement of the oldeft life by 6 times the amount of 11. for one year; 2d. fubtract the quotient from the value of the oldeft life; 3d. multiply the remainder by the complement of the oldeft divided by twice the complement of the youngeft; 4th. add this product to the value of the youngest life, and the fum will be the value of the longeft of the two lives.

QUEST. What is the value of an annuity of 11. to continue during the life of the longest liver of two perfons, whofe ages are 43 and 54; allowing 4 per cent. compound intereft?

86 86

1.04

6.24)32.00(5.128 3120

800

624

5.128

To find the prefent value of an annuity for the longest of three lives; in this is four cafes.

Ift CASE. When all three lives are equal.

RULE. ft. Divide the complement of a fingle life by 6 times the amount of 11. for one year; 2d. fubtract the quotient from the value. of a fingle life; 3d. take of the remainder and add it to the value of a fingle life, the fum is the anfwer.

QUEST. Let the three perfons be each 54 years old, and compound intereft be allow'd at 4 per cent. what is the prefent value of the longest liver?

2d. CASE. When the two youngest are equal.

RULE. ft. Divide the complement of one of the youngest lives by 6 times the amount of 11. for one year; 2d. fubtract the quotient from the value of one of the youngest lives, and take half the remainder and call it A; 3d. divide the complement of the eldest life by 6 times the amount of 11. for one year; 4th. fubtract this quotient from the value of the eldest life; 5th. multiply the remainder by the fquare of the complement of the eldeft life, and divide by 4 times the fquare of the complement of one of the youngest lives; add this, and A, and the value of one of the youngest lives together, the fum is the value required.

QUEST. Let the two youngest be 34 and the eldest 54, what is the value of an annuity for the longeft of the three lives, at 4 per cent. ?

See the remainder of the work in the next pages

3d. CASE. When the two eldeft are equal.

RULE. Ift. Divide the complement of the oldeft fingle life by 6 times the amount of r1. for one year; 2d. fubtract the quotient from the value of the oldeft fingle life; 3d. multiply the remainder by 3 times the complement of the oldest fingle life, and divide by 4 times the complement of the youngest life; 4th. add this laft number to the value of the youngest fingle life, the fum is the value of the longeft of the three lives.

QUEST. What is the value of an annuity at 4 per cent. for the longeft of 3 lives, the youngest is 20, and the two oldest are each 34?

See the work in the next page.