328 METHODS OF INTERPOLATION. 



probable value of the correspouding fourth difference, so that if the 

 latter is a minimum, the former will be a minimum also. 



In connection with the least-square formulas, given at top of page 327 

 of the previous memoir, it was stated that those formulas render the 

 probable error of the adjusted term a minimum, and this property was 

 also claimed for them by Schiaparelli, who had obtained them earlier, 

 in a different way, as mentioned on j)age 335. Bat these conclusions 

 w^ere reached by estimating the probable error £o of the adjusted series 

 in a manner which was really defective, since it did not take any account 

 of the condition of continuity which that series ought to fulfill. We 

 shall see hereafter how the probable error may be estimated on the 

 other, and, as it seems, the more correct principle. 



The formulas (69), (71), &c., being definitely fixed upon as the best, it 

 became desirable to extend them so as to include as many terms as 

 possible, and they have been computed accordingly, for 17, 19, and 21 

 terms, as follows: 



«9 = 167X525 { 334323 «9+308308 (?^8+«io)+ 238238 {th+Un) \ 



+ 14523o(«c+«i2)+56056(?(5+M,3)-6664(«4+«i4) (85) 



-32844 (^/34.i/,i5)_28424('W2+Wi6)-11305 (Wi+^^it) \ ) 



Uw = 42010995 { 7^27477 «io+ 7135128 (m9+«(:^) + 5783778 («8+Mi2) 



, +3913728 («,+«!.) + 1979208 (W6+«i4) + 411264 [ih+i^^) 



— 515508 {Ui + Uw) — 767448 [u^ + Wn)— 549423 («. + u^^) 



-198968 (Wi + *fi9) I 



(86) 



«n = 7^^^^^ |l2739727«n +12046320^i.(io+«i2) + 10115820(«g+«i3)\ 



+ 7360320 (Wa+^^l4) + 4351320 (m,+%5) + 1674432 (m6+Wic)( 

 -222870 (^5+ Wit) -1149120 (W4+«]8)- 1213245 («3+«i9)f ^^'' 



• 



■769120 (w9+«2o) -259578 (Wi+«2i) \ 



The work of computation need never be repeated, for the accuracy of 

 the formulas can be readily tested by the condition that the weights of 

 any one of them, taken together with the eight nearest zero weights, 

 constitute a series of the tenth order. For practical use, it will be most 

 convenient to i)ut the weights in decimal form, and to retain only three 

 places of decimals. These are given, for the whole series of formulas, 

 in Table III. 



