344 METHODS OF INTERPOLATION. 



four consecutive birthdays, and let us assume that they are connected 

 with the age by the law — 



;, = _ _^ = A + B .T + C a;-2 + D ^3 

 ydx 



Placing the origin of coordinates at the middle, and assigning to x in 



succession the values — f ? — i, + -g^, and + |, we have — 



/^-i = A-f B + f C--y-D 



/.2 = A-iB + iO-i D 



/.3-A + 1.B + 1O + 1 D 



/^-4 = A + f B + f + \M) 

 from which the four constants can be found, and we have — 



A = tV |9 (,«2 + P-^) - {!H + ,^4)1 



But integration gives us — , • 



— log' 2/ = A ^ + i B aj2 _|_ ^ Q ^3 _j_ 1 J) ^.4 _f_ constant. 

 At the second birthday, taking x = — ^ and y = 2/2, we have — 



- log' 2/2 = - 1 A + i B - J^ C + gL B + constant; 

 and at the third birthday, with x — ^ and y = 2/3, we have — 

 - log' 2/3 = L A + i B + 2^ C + J^ D + constant. 

 Subtraction of the first of these two equations from the second gives — 



-lo§'(|) = A + J,0 



But — is the probability j92 of living one year from the second birthday; 

 so that, substituting the values of A and 0, we have — 



- log' V2 = 2V 1 13 (/.2 + !,s) - {!J.i + !h) \ (100) 



and this is the formula sought. If, for example, ,ai, /-/.2» &c., denote the 

 .intensities at the ages 39, 40, 41, and 42, then log' p^ will be the Napier- 

 ian logarithm of the probability that a person aged 40 will live at least 

 one year, and 1 —1)2 is the probability of dying within the year. 



INTERPOLATION BY MEANS OF A CIRCULAR FUNCTION. 



We have already noticed that the algebraic formulas (A), (B), (C), &c. 

 of the previous memoir can be employed in place of the usual formula of 

 finite differences, for making ordinary interpolations from single terms 

 taken as data. In a similar way, the circular formulas (a), (ft), (c), &c., 

 given at page 336 of the memoir, are capable of being used for the i>urpose 

 of ordinary interpolation, and may thus take the place of sucb formulas 

 as are found in the appendix to vol. II of Dove's Repertorium, page 

 1575. The given single terms are supposed to be either three, four, six, 

 •eight, or twelve in number, and to be situated at equal intervals through- 

 out the whole circular period. We have only to regard them as consec- 

 utive areas represented by Si, S2, &c., respectively, and take n equal to 

 unity. For example, let there be four of these given terms or equidis- 

 tant values of the function, namely, 



7, 12, 5, 9 



