METHODS OF INTERPOLATION. 343 



covering tbe Irue law is chimerical, as there seems much reason to think, 

 it may be doubted whether the quantity [j. can have much practical 

 importance. We will proceed, however, to obtain some formulas for lind- 

 iug the intensity, which cannot be directly observed, from the observed 

 quantities; and, conversely, for finding the latter from given values of 

 the intensity. 



Let us consider any five consecutive birthdays, and let the numbers 

 of persons annually reaching them be — 



2/b 2/2, 2/3, ?/4, ?/5 



We will assume that the relation of these numbers to the ages may be 

 regarded, within the limits of these five years, as being of algebraic 

 form and of a degree not higher than the third, so that we have — 



y = A+ B X -\- x^ -[-J) x^ 

 We will also suppose, for simplicity, that the age x is reckoned from the 

 middle birthday, taken as an origin of co-ordinates. Then assigning to 

 V in succession the values —2, —1, 0, +1, and +2, we have — 



2/i = A-2B + 4C-8D 



2/2 = A-B + C-D 



3/3 = A 



2/4=A+B + C + D 



2/5 = A+2B + 4C+8D 



and these equations determine the values of the constants. We find — 



A = 2/3, B= -i-L 18(2/4 -2/2) -(2/5 -2/1) ( 



But when x = 0, we have y =A. and ^ = B, so that the intensity of 

 mortality at the middle birthday is — 



and consequently — 



a - ^2/ _ B 



y ax A 



,,= ^^y'^-y^l-^y^-y^) (99) 



which is the formula sought. Thus, if we have found by observation 

 the numbers of persons annually attaining any five consecutive birth- 

 days, or, what amounts to the same thing, the "numbers living" at 

 those five ages out of a. given number of persons born, the above for- 

 mula will give the intensity of mortality at the middle birthday. It is 

 a little more accurate than the usual formula, 



^?/2 



The numerical value of p. does not differ very greatly from that of the^ 

 probability of dying within a year. 



We turn now^ to the converse problem, that of finding the probability 

 of living one year, or of dying within a year, from certain given values, 

 of the intensity of mortality. Let ,ai, ,v-2, !J-2i ih-, be the intensities at any 



