334 METHODS OF INTERPOLATION. 



and when each term is supposed to be affected by a probable error or 

 deviation e, the probable value of J4 becomes — 



+ (A;'-22/j+100y^+(45-8Af+(10-/.f+l] 

 which reduces to — 



W= j.> ,Qj._.,fy J'iOk' 2+4302 />;'^-1008A-/i'+4004 k'-S58i)(i /c+75476 



Eegardiiig (J4) as a function of the two independent variables fc and k', 

 we have the two equations — ^ 



dk ~~ ' dk'~ 



giving the values A-=:fi and A:'=^^, which render (J4) a minimum. 

 Substituting these in (70), we get the adjustment formula sought— 



«, = 3-^v^[469«4+324(«3+«5) + 54(«2+«6)-60(«i+'»,)] . . (71) 



It is found that in each of the formulas (09) and (71), the whole series 

 of weights, taken together with the eight nearest zero weights, consti- 

 tutes a series of the tenth order. By means of this property, we can 

 construct with greater fiicility the following similar formulas : 



«5 = 83W[-S84^'5 + 22G8(»4 + Wg) + 91S(K3+"7)-132(«2+»8) 



-297(^(i+«9)] 



«C = ^5l94[7308l<6 + 61G0(W5+«7) + 3410(»4+»8) + GG0(«3+"9 



— 71'){ir,+nio)—512{Ui-\-i(n)] , 



[(72) 

 (73) 



„,= _j__[48G36wT+427C8(«6+«,)+27918(«5+«9) + 108G8(«4+«io)r 



-1287(«3+«ii)-5148(?/2+?/i,)-28G0(«,+ «i3)] V 



«8=W45o[827<54w84-74844(»,+«y)+54054(?/„+Wio)+2S028(«,4-«„) ) „ 

 + 5733(»4+«i2)-6552(W3+«i3)— ^092(?<,+ Wi4)-3G72(?/i+?fi5)] ) 



If the smallnoss of the fourth differences of the adjusted series is to be 

 taken as the ultimate and only test of its regularity of curvature, it will 

 follow that these formulas ought to be used in preference to (53), (54), 

 (55), &c., from M'hich, indeed, tliey do not differ greatly, as can be seen 

 on comparing their decimal weights. The probable errors of the ad- 

 lusted terms, however, are increased a little, and the weights follow a 

 curve which is not precisely tangent to the line of the zero weights. 



At all events, the same principles can be usefully employed in fixing 

 the weight of the middle term in ibrmula (48), so as to give greater reg- 

 ularity to the adjustment of a double series. By a process precisely 

 analogous to that by which (G9) was obtained, it can be proved that in 

 order to render the probable value of the yomplete second difference 



