METHODS OF INTERPOLATION. 323 



are to form a series of the sixth order, their seventh differences will be 

 zero, giving the equation — 



_5_7(8_7.:) + 21-(13 + 4/0-35(13+47.-+A:') 



+35(13+4A;)-21(8-fc)-35=0 



Also, since there is to be a maximum at the initial term 0, the differ- 

 ences of the series of weights must satisfy the condition — 



giving the equation — 



1800 + 460(8-A;)-472(134-4A:)+225(13-4-4A-+7/)=0 



We have then two equations, from which the numbers 1c and k' are as- 

 certained to be — 



7^ 2 03 6 7./ fi679 



SO that the adjustment formula becomes — 



«4=55yo4[-3104M4+1042;j(M.3 + "5)+3000(??3 + ?^G)-3185(^^:+»,)] . (52) 



Here the nine weights — 



0, -3185, 3060, 1G425, 23104, 1G425, 30G0, -3185, 



form a series of the sixth order, and if their successive orders of differ- 

 ences are taken they will be found to satisfy the equation (51). The 

 following formulas, comprising five, nine, and eleven terms respectively, 

 possess properties similar to the above : 



^5=58iT3[19375«5+1569G(«4+«c) + 7050(«3-fW7) 



-704(«24-«8)-2349(«i+«9)] 



% =¥TyTmr2oT2T8 [^^•^*^719420G «64- 51593437700(«5+ W7) 



■+-3131529GG40(«/,4-l-?<8)+82778GGG85(?f3+«9) 

 -G224G58450(«2+Wio)-60704355G9(«i+ifii)] 



It will be more convenient in practice to have the weights expressed by 

 decimals, as follows : 



W3=.5GGlG«3+-28923(«2+«4)-.07231(?Aj + ??r.) • • (o3) 

 W4=.41476?<4+.2948G(H3+«5) + .05494(?^3-f?fo)-.0571S(?/i-f^^) . . (54) 



W5=.329GGw5-f .2G706(w4+«G) + .l-00G(w3-f«7) } ^_ 

 -.01198(«2+Wa)-.03997(«i-f«9) ^'^''''^^ 



W6=.2740G «e + .23737(«5+«7) + .14408(?^4+7^8) + .03809(«3+?^c,) ^ ^ 

 -.02864(%-f?/io)-.02793(wi+«ii) ^^''"^'^ 



Without attempting solutions in whole numbers, we can proceed in a 



