METHODS OF INTERPOLATION. dz\) 



This formula may be used advantageously in constructing a graduated 

 rate of mortality similar to series (h) in Table II. The simplest mode of 

 procedure will be to obtain the equation of the graduated series of the 



form — 



u=A' + B'jc+OU^-\- +1'*^ 



and to compute by logarithms first the values of B'.r for all the ages, 

 then the values ofC'j?^ in like manner, and so on, and finally to take the 

 aggregate of the values at each age. The accuracy of the work will be 

 tested by the condition that the sums of the terms in the corresponding 

 groups in the graduated series must be severally equal to those in the 

 given one. It should be also mentioned that, to insure accuracy, the 

 multiplications within the brackets in formula (G), such, for instance, as 

 that of S5 by its coefficient 11702134, &c., ought to be performed arith- 

 metically and not by logarithms. 



INTERPOLATION BY MEANS OF AN EXPONENTIAL FUNCTION. 



When values in arithmetical progression are assigned to x in the ex- 

 ponential equation — 



i/=Z</5^+o^"+fZo^'+ &c. 



the resulting values of y will be terms in a recurring series, whose order 

 is denoted by the number of constants ft, y, (5, &c. The above formula 

 has sometimes been used for the i>iirpose of ordinary interpolation, and 

 represents a curve which, under certain conditions, can be made to pass 

 through any number of given points whose ordinates 2/0? Vu 2/2? &c., are 

 equidistant. The whole number of constants h, c, d, /?, ;-, S, «&c., included 

 by the formula, must be equal to the number of points given. If this 

 is an odd number, we must write — 



y=a-^h l3''-{-c y''-\-d S"-^ &c. 



For the most general method of determining the values of the constants 

 in any given case, see articles by Prony, in Vols. I and II of the Journal 

 de VJ^cole Poli/techniquc. We may here remark that if there are not 

 more than five constants, their values can easily be obtained in the ordi- 

 nary way, first eliminating a, h, and c from the equations of condition, 

 ,then finding the values of /5 and y, and afterward finding those of a, &, 

 and c. 



Now let us write the general equation under the form — 



2/=A+(Blog'/S)/5'+(ClogV)/''+(Dlog'(J)o^+&c. . . (G5) 



where log' denotes the Naperian logarithm. Integrating ydx between 

 the limits x—^n and x-\-^n, we get — 



S=A n+B(/Si«— i?-i")/5a;+C(^i"— /'-i")r''4-D('J*" — (J-i'i)o^+ &c. (66) 



which is identical in form with the expression for y, so far as the abscissa 

 X is concerned. Consequently, if we assume a series of groups contain- 



