METHODS OF INTERPOLATION. 321 



and (47) becomes — 



W4 + ?/5+W6=3(A+-rVD + fE) 



Likewise the rectaugle cc' gives — 



W7+«84-"9=3(A-B+ifD + fE) 

 Again, for the rectangle adf we have — 



S=?^i + »2 + W4-f ?<5+W7+?^3, w=3, «=2, a'=0, y=\ 

 and (47) reduces to — 



In like manner the rectangle dc' gives — 



«2+W3+«5+"G+"8+»9=r>(A-iC + aD+j^E) 



We have thus obtained five equations by wliich to determine the five 

 constants A, B, C, D, E, in terms of the tal)ulated values Wi, Mjj ^'3, &c. 

 Now, in the middle one of the nine divisions we have — 



S=«<5, wi=l, «=1, j==0, y=o 



and formula (47) becomes — 



^/5=A+J^D+tVE 

 Substituting in this the values of the constants A, D, and E, we arrive 

 at the result — 



M5=-1-[5W5 + 2(«2+?^4+''g+"8) — (''l + W3 + «7+"9)] • • (48) 



and this is the adjustment formula required. Its accuracy can easily be 

 tested by trial with any table constructed from an equation of the form — 



%= A' + B'.r+ C'y + D'.i'2_^E'/ 

 the adjusted value being in this case the same as the original one. In- 

 deed, we shall find that the result is exact, even when the table has been 

 constructed from a complete equation of the third degree. 



Again, to adjust the value of a term occupying the middle of one side 

 of the assumed rectangle, as Ui^ for instance, we have — 



S=W2j m=l, M=l, a;=l, y=^ 



and consequently — 



?<o=A+B + if D+J^E 



Substituting the values of A, B, D, and E, we obtain the adjustment 

 formula — 



W2 = i[5«2 + 2(?<l + «3 + ?/5 + W8)-(?^4+«c4-«7 + W9)] • • (49) 



In a similar way the adjusted value of a term like Mi, occujiying one 

 corner of the assumed rectangle, is found to be — 



Wl=M3Wl+2(M2+«3 + W4 + «7)-(W5 + «C + W8 + «9)] . . (50) 



By one or other of the three formulas here given, the value of any term 

 in an irregular table can be approximately adjusted, and, as in the case of 

 an ordinary series, the weight of the term to be adjusted may be in- 

 creased or diminished at pleasure. 

 21s 71 



