different Tables of Mortality. 353 



or, since A=l_2i, A=X_1 (3hL. + ^ - r-* 

 a k \ CO cc 



— + I j: that is, at 4 per cent, and taking c = 88.5, and 



c cc J/ 



JL = .0113, X being == 25.497, XX = 650, and ~ = .^8812, 

 c 



A = 25.497 - 2.(4.233 + 'I66q - 18. 923 r*"')- 

 k 



K. Annuities at 4 per cent, from the quadratic hypothesis. 



Mean — .04 

 ft is obvious that a mean error so small and so subdivided is 

 as likely to belong, in great measure, to the observations as to the 

 computations. 



23. It was my intention to proceed, in a similar manner, through 

 the computations of annuities on two joint lives, and of the con- 

 tingent reversions of survivorships : but the accuracy of the pro- 

 posed formulas appears to be already abundantly demonstrated by 

 the two last comparisons, and I shall confine myself, for the present, 

 to the great remaining problem of three joint lives^ the facilitation 

 of which would be really a step of practical importance, even if we 

 allow.ed the accuracy of the existing tables, which have been the 

 most extensively employed for calculations of this kind. 



24. The age of the eldest of three lives being ar, and the ages of 

 the two younger x — p' and x — p'^ the initial values of s, s', and 

 s'' being k, k\ and A/' respectively, when a; = ^; the probability 

 of the survivorship of the whole three, for any other values of a:, 



ss's' 

 will be — — -, and the fluxion of the present value of the annuity 

 kkk' • 



