354 Practical Comparison of 



will be ii^ r'-^ dr. Now, 5 = 1 - ^, s' ^ 1 - ^fZ^). , 

 kkk cc CO 



and s/' :=. \ — .^: ^-J- , or neglecting c, which always accom- 



cc 



panies x and p^ till the end of the computation, s' r:^ s + 2j!?'',r — p'p\ 

 and s'^ = s + 2j9"a; •— pf" \ whence ss' =ss + 2p'a;s — jp^p'.s, and 

 ssV = s* + 2/a^s^ - p'^V + "^pxi + 4j9'j9Vs -- 2p'p'p''ocs - 



y^v - 2p'py'xs + p'p'pVs = s« - (j^y + p'y) «' + 



p'p'p"p^fs + 2 (^' + yO s'.r + V2?'^s^' - 2j9'p" {p' + p") s^. 



25. Hence it appears that this contingency comprehends that 

 which belongs to the value of three equal joint lives, as well as those 

 which relate to one life and to two, these latter being also compli- 

 cated with the expression of the age and of its square : and it is 

 obvious that the result may be reduced to the form kk'k"A = Q — 

 ipV + pV) ^ + p'p'pYQ'' + 2(p' + p") Q'f + Ap'p'^Q'"'-^ 

 2p'p" (:p' -i-p") (^"^' '. all the quantities Q. . . being dependent on 

 the oldest life only, and capable of being expressed in a table by as 

 many single numbers for each age, to be afterwards combined 

 according to the variations of the younger lives, as here expressed by 

 the differences. The first three numbers might be readily obtained 

 according to any given tables of observations from the tables of the 

 values of ecpial joint lives already in existence, or they may be com- 

 puted with the rest, from the formulas : the last of all is also sub- 

 servient to the calculation of the value of survivorships. 



26. In the first place, for the quantity Q, belonging to three equal 

 lives, we have -/sV^-Mjt = — /r^-^do; (1 - 3^^ + ^x^ - x^) ~ 

 ^r'^ (1 — 3 {x^ + "l-hx + 2a^) + 3 (:i-4 + 4Aa?3 -f 12aV + 24^3^;* + 

 24X'') - {3p + 6^a;5 + 30aV -f \2^>?x? + 360A''a.-' + VZ^-k^'x + 

 720^^) ; which, taken from x t^:: q to a? = c, becomes A (1 — 3 

 (g* + 2^9 + 2ax) + 3 ((?" + 4Ag3 + i2aV + ^^^^^Q + 24x'») - 

 (g« + 6xq^ + 30A«g4 + l20xY + seOx'^q^ + 120\'q + 720^^) - 

 r'-' [1 -3 (1 + 2a -1- 2xx) + 3 (1 + 4a + 12a' + 24a^ + 24aO - 

 (1 + 6a + 30a* + 120A3 + 360A4 + 720a5 + 720a*')]) = Q : the 

 numbers A and g, wherever they occur after the first A, being under- 

 stood as divided by c. . 



