326 METHODS OF INTERPOLATION. 



small number of consecutive terms in it will be approximately of an 

 order not higher than the third. Hence, if any series, such as a table 

 of mortality, is thus adjusted, its fourth differences will be small, and 

 positive and negative values will bo equally probable. 



In the case of formulas like (22), which hold good for a series of the 

 fifth or any lower order, we may fix the local weights of the terms by 

 these two conditions, that the whole series of weights, including the 

 two nearest zero weights, should be of the eighth order, and that it 

 should have minima at the beginning and end, so as to satisfy the equa- 

 tion — 



J,-$+4'- -t=o 



Jo o 



Thus we obtain the formula — 



«4=^3j1^^[7008m4+307o(m3+M5)-1470(»2+"g) + 245(»i+«,)] 

 which, with decimal weights, is — 

 «4=.61922«4+.28559(«3+«5)-.11424(?/,+«6) + .01004(«i+«,) . . . (59)* 



To find formulas for adjusting the first two and last two terms of a 

 series, we may proceed as follows : Assuming that five terms, Wi, i<2, «3, 

 «4, %, form a series of the second order approximately, and taking the 

 equation — 



with the origin of coordinates at the middle term u^, we have the five 

 equations of condition — 



«i=A-2B+4C 



1/3= A 



W4=A+B+C 



«5=A+2B+4C 



Combining these by the rule of least squares, we find that the values 

 of the three constants are — 



A = 3V[17w3+12(«2+«4)-3(«i+K:,)] 



B=j-V[2(W5-%) + («4-«2)] 



C=tL[2(Wi+M5)-2w3-(??2+M4)] 



and consequently we have — 



Wi=J5(31mi+9»2— 3«3-5m4+3w5) . . . (CO) 



i/2=JL(9«i+13w2+12j/3+Gyf4-5i/5) • • • (01) 



which can be used with advantage in place of (25) and (26), if the series 



* This formula may be used wbeu the law of a giveu series varies so rapidly that five 

 consecutive terms cannot be regarded as formiug a series of an order not higher than 

 the third. 



