362 - KEPOET— 1880. 



+ ^ (x^ A2 + 2xy AS + f 32) «„ o 



+ 



Bnt this supposition is entirely arbitrary, and not even justifiable if it 

 happens to be known tbat the second and higher differences are con- 

 siderably greater in one direction than in the other. This, and the remedy 

 for it, are only to be ascertained by a special study of the function. 



In accordance with common practice, the direct expansion by ordinary 

 differences has been used; but there is no motive for this except con- 

 venience and simplicity. Subject to the difficulties of interpretation and 

 handling which they introduce, any form of the expansion 



«,, = (1 + A)- (1 + A)" uo = E'^ E" uo 



•will answer the purpose. There are even cases in which it would be 

 desirable to form an interpolated table for the given value of one variable 

 on the supposition that the other is constant, and then to interpolate 

 that as a single entry table. 



Thus, supposing that going along a line in the table represents the 

 variation of y, and going down a column represents the variation of x, 

 then the whole column corresponding to a particular value of y might be 

 interpolated, value by value, so as to give the series arranged in 

 column 



and then interpolation to w^,,, might be effected by single entry ; or the 

 process might be reversed, and the line 



■WjO '*.ll '^3:2 ^^.3 



formed first, and then v,„ by single entry interpolation. 



It is worth while to remark that the interpolation may occasionally 

 be reduced to single entry interpolation along a diagonal line. This is 

 always the case in bisection ; but it is by no means confined to that 

 case. 



The inverse problem of interpolation is in general of a higher order of 

 indeterminateness, unless some other data are given than the mere value 

 of the ordinate. For if the two variables be regarded as horizontal, and 

 the function as representing a vertical ordinate, then the ordinate being 

 given in value, merely furnishes a locus, namely, a level line. So if one 

 of the variables be given, a special table may be formed for that value of 

 the variable, and then the inverse interpolation is an operation of single 

 entry. . But if, instead of this, some equation be given between x and y, 

 the problem falls out of the province of direct synthesis, unless the 

 relation between x and y be that they are in a constant ratio. The 

 analogy is of course that of the representation of a surface in geometry of 

 three dimensions, and that analogy is really the key to the question. 



An example of the interpolation of a double entry table as far as 

 third differences is given by Legendre.* 



Quadrature in two dimensions is really equivalent to the evaluation of 

 solid volume. The two quadratures may be effected either separately, 

 or by the methods indicated in Section IV, 4, of this report. 



Another problem arising out of a double entry table is that of inter- 



* Traitc deg Fonctions Elliptiqves, vol. ii. p. 201 (cap. xv.) 



