METHODS OF INTERPOLATION. 333 



(See the figure 0(1 page 320 of the previous memoir.) By a process 

 strictly analogous to that which we have followed in the case of ordi- 

 nary series, it can be shown that the probable error of a term in the 

 double series bears a fixed ratio to the probable value of the correspond- 

 ing complete second difference, namely, 



that the formula (76), which renders the probable value of J2+2 in the 

 adjusted series a minimum, is, therefore, the one which will make the 

 most accurate adjustment; and that the probable errors of the adjusted 

 and unadjusted terms will, in this case, bear to each other the ratio — 



i'= 0.305 



£ 



If the least-square formula (48) were used, the ratio of error would be 

 increased to — 



- = 0.460 



£ 



It was remarked in the previous memoir that formula (48) gives exact 

 results when applied to a double series or table constructed from a com- 

 plete equation of the third degree. This is also true of (76), and, more- 

 over, such formulas will be found exact in the case of a table constructed 

 from an entire i)olynomial of auy degree whatever in x and y, provided 

 that no term of this i)olynomial shall contain x^ y^ as a factor. This is 

 only a particular case under the general theorem that if we denote by 

 Juj+n the result of m finite differentiations with respect to x, and n with 

 respect to y, we shall always have — 



in a double series constructed from an entire polynomial of any degree 

 whatever in x and y, provided that no term of this polynomial shall con- 

 tain x^ y^ as a factor. 



TEST OF A GOOD ADJUSTMENT. 



When a table of mortality or other irregular series is to be adjusted, 

 for instance, by one of the formulas of Table III, the question presents 

 itself, Which formula is most suitable for use in the given case? This 

 question cannot be answered definitely in advance, but we can easily 

 see that if only the five-term formula is used, the adjustment may be 

 insufficient ', that is, some undulations may be retained which are due 

 to accidental causes, and do not properly belong to the true law of the 

 series. On the other hand, a formula of twenty-one or more terms 

 might smooth out the series too much, so as to obliterate some features 

 of the natural law. The necessary course of procedure will be, to judge 

 as well as we can which formula is most likely to be suitable, and, hav- 

 ing adjusted the series by it, to apply some test of its sufficiency. The 

 natural test seems to be this : that on taking the differences between 

 the adjusted and unadjusted terms throughout the series, the system 



