320 



METHODS OF INTERPOLATION. 



«, X, and y, so that every such group furnishes an equation of condition 

 which, besides the constants A, B, C, &c., contains only numerical 

 quantities. By assuming as many groups as there are constants, we 

 shall always be able to tind numerical values for the constants, and sub- 

 stituting them in formula (40), and nmking — 



m = l, 7i=l, S=w 



•we shall have an equation of the form — 



u=A' + B'x-\-C'y+D'j;'~-\-Wf+F'xy+&c. 



which will be the equation of the graduated table, and from which that 

 table can be constructed by assigning to x and y the proper series of 

 values differing from each other by unity, so that they shall represent 

 in succession the coordinates of the middle point of each division of 

 the table. 



We can also make an approximate adjustment of a double series by 

 formulas analogous to those which we have already found under the 

 second method for adjusting an ordinary series. For exami^le, any nine 



adjacent terms i<i, ti2, th, th being grouped in a rectangle with three 



c' h' a' 



X 



terms on each side, as in the figure, let it be required to find a formula 

 by which to adjust the value of the middle term u^. Let us suppose that 

 the ecpiation of the curved surface is — 



;?=A+B.r+C2/-f D.r2+E7/2 



then F aiul all the succeeding constants disappear, and formula (40) 

 becomes — 



Now, in the rectangle na' we have — 



S=?<i+2f2+«3, m=l, ?<=3, a'=l, y=0 

 so that (47) reduces to — 



«i + »,+ n3=3(A+B + ifD+fE) 



So, too, in the rectangle hh' we have — 



S=j/4 + «5+?^G. w = l, w=3, .37=0, y=0 



