352 



METHODS OF INTERPOLATION. 



.3he constant interval between the weights be taken as the unit of x, 

 Schiaparelli's two conditions give the two equations — 



23 A + 2 { (IP) B + (114) c + (iiG) D + (11«) E + {IW) F} = 1 

 (IP) A + (ll-*) B 4- (IP) C + ill') D 4- (ir«) E + (1P2) F :^ 

 where the notation used is — 



(11") = 1™ + 2°^ + S'" + + 11"^ 



Since the curve passes through the positions of the eight nearest zero- 

 weights, we have also these four equations : 



A + 122 B + 124 C + 126 1) + 128 E + 1210 F = 



A + 132 B + 134 C + IS'' D + 13» E + 131" F = 



A 4- 142 B + 144 C + 14« D + 143 B + 14i« F = 



A + 152 B + 154 C + 15" D + 158 E + 1510 F = 



This makes six equations in all, enabhng us to compute the numerical 



values of the six constants, A,,B, &c., in the equation of the curve, after 



which, by assigning to x in succession the values 0, 1, 2, 3, &c., we get 



values of y which are the weights of the adjustment-formula sought. 



Commencing with the middle term, they are — 



.1534 .1463 .1262 .0969 .0635 .0317 .0065) .,^^. 

 —.0092 -.0151 -.0133 -.0077 -.0025) ^^^^' 



In a similar way, the weights of the 25-terra formula have been found 

 to be — 

 .1424 .1368 .1205 .0962 .0678 .0395 .0151) .^a, 

 _.0{)24 —.0119 —.0140 —.0110 —.0080 —.0018 \ ^^""^^ 



If it were desirable, there would be no great difficulty in extending 

 the series of formulas still farther. 



In each of the foregoing cases, if the greatest nicety of adjustment is 

 required, the weights ought to include as many as four places of deci- 



mals, because, if the fourth figure is neglected, the error-ratios — are 



found to be materially increased. These ratios, as computed by formula 

 (92), are respectively .00124 and .00091 when four places are retained, 

 but with only three places they are increased to .00191 and .00197 re- 

 spectively. 



On the other hand, it is found that the error- ratios for the 5, 7, and 

 9-term formulas of Table III are not materially altered by dropping the 

 third decimal place, so that the first part of that table might as well 

 stand thus : 





5 



7 



9 



7o 



.56 



.42 



.34 



h 



.29 



.29 



.27 



h 



-.07 



.05 



.11 



.h 





-.05 



-.02 



u 







-.03 



e_ 



.23 



.080 



.038 



e 









