different Tables of Mortality, 855 



!Sl7. The second quantity is Q' e= — /r'-« ««dx, is being = 1 - 

 2 ff + fl , or 1 - 2a:a: + or* ; which gives the fluent Xr'-» (1 —2 



(a;* + 2aj? + 2^^) + a:< + 4\r» + 12xV + 24x'a;+24x*) : and this 

 from a? = g to a: =: c is ^ (1 - 2<f — 4Agf — 4^x + g* + 4^5^ + 

 12aV + 24x'g + 24a* — r'-' [1 - 2 — 4x - 4ax + 1 + 4a + 

 12a* + 24a'' + 24a*]) = Q' : restoring c in its place. 



28. For Q' we have — fr^'^sAx, as in the case of a simple 

 annuity (22), = Ar*-* (1 - jc' - 2at - 2a«) giving A (1 - 7* — 

 2Ag - 2AA + f '-^ (2a + 2A«) = A (A; - 2A5 - 2AA + 2Ar'^-' 



[1+^]). 



29. In the next place Q'" =: - fr'-'^ssxdiX r= ^ fr'"'^ {xdix - 

 %x^Ax + a;Mx) = Ar'"' (jf + A - 2 (a;^ + 3Aa?' + Qk\v + G>?) + 

 x* + 5Aa;* + 20AV + 60aV + 120A*a? + 120a*) ; which, from x:=iq 

 to x=c, gives A (g + A - 2(g' + SAg^ + 6X«g + 6a^) + 

 ^5 + 5xg* + 20aY + 60aY + 120A*qf + 120A« - j-^"' [1 + a - 

 2 ( 1 + 3a + 6a« + 6a^) + 1 + 5a + 20a« + 60a^ + 120a* + 

 1:20a']). 



30. For Q"", derived from sx\ we have — /r*"' (a;Ma? - j?*dar) = 

 xr*-* (x« + 2Aa; + 2aa - [a;* + 4Aa;« + 12aV + 24A^jr + 24a*]) ; 

 and this, when corrected, becomes A (jf + 2'Kq + 2aa — 5* — 4A9' — 

 12aV - 24a'9 - 24X* - r'-' [1 + 2a + 2aa — I - 4a - 

 12a« - 24'A - 24a*]. 



31. Lastly, for q['"f, from sx, we have - fr'"' {x — x^) dx = 

 Ar*^ (a? 4- A - (a:* + 3Aa;' + 6A'a; + 6a'), which becomes \ (q + 

 X - (9« + 3\q^ + 6\\ + 6\») — r<^« (1 + X- - [1 + 3\ + 



32. Taking, for a single example, the value of three joint lives of 

 30, at five per cent., we have ^ = 80, J- = . 01 13, ^ = . 339, \ = 



20. 5, A = . 23165, and r""' = . 0576. By salstituting these 

 c 



quantities in the expression _ = A, we have the value 11 .37. 



