METHODS OF INTERPOLATION. 281 



the population liviDo- between the ages 43i and 44^, 44;} and 45i, 45} 

 and 40}, &c., respectively, aiid a comparison of formulas (.3) and (.")) 

 sliows that the two sets of numbers would be almost identical, though 

 not precisely so. The difference between them is — 



y — « = 2 4 ko (- ^2 — Si — S3) 



a number so small that it will not ordinarily affect the first five signifif- 

 cant figures of a result. 



A considerably larger error is involved in the assumption that the 

 ratio of the deaths annually occurring within any decade of age to the 

 population living within such decade represents the annual rate of mor- 

 tality at the exact middle age of the decade. ( Assur. Mag.,Yol. IX, p. 125.) 



Let Si, S2, S3, be the deaths, and Si, S2, S3, the population, for any three 

 consecutive decades, then the deaths annually occurring at the exact 

 middle age of the middle decade are, by formula (5), making x—0, 



and the population living at the same age is, 



Yf?-''=oio [2G S., — (Si+S3)]^7.c 

 so that the annual rate of mortality at that exact age is, 



y 2G«.3 — (.S, + .S;;) 



Y i'G82— (S1 + S3) 



(G) 



The difl'erence between this and the assumed value A is sufficient to 



S3 



alter the fourth significant figure of the quotient, and even the second 

 and third at the older ages, as can easily be verified by assigning to Si, 

 Si, &c., the numerical values for the various decades given by their log- 

 arithms in Table III of the Preface to the English Life Tables. 



As regards the general accuracy of interpolations made b}' formula 

 (2), it must be noted that near the middle point of the middle interval 

 ?«2 the values obtained will be more accurate than they will be at its 

 extremities, and the accuracy attainable will diminish as we proceed out 

 of the middle interval into either of the lateral ones. This is analogous 

 to what we know to be the case with ordinary interpolations by second 

 differences. And just as the degree of accuracy is increased by taking 

 third differences into account, so here we can increase it by taking four 

 intervals instead of three. This will give a curve of the third degree, 

 which admits a point of inflexion, and is, therefore, better adapted than 

 the common parabola to represent the form of a series whose second 

 difference changes its sign. 



For thesakeof simplicity, let us assume that the four areas Si, S2, S3, 

 S4, are symmetrically arranged with respect to the axis of Y, so that 

 the distances from the middle ordinates of Si and S4 to that axis shall 

 be each equal to «], and the corresponding distances for S2 and S3 each 

 equal to «2, while the bases of the first and fourth areas are each equal 

 to ??), and those of the second and third are each equal to ih- Then taking 

 the curve — 



