ON QUADHATDRES AND INTERPOLATION. 361 



The tables given by Cotes and Woolhouse for symmetrical diiFerences 

 appear to have been formed upon the same principle. 



A question of some interest in the interpolation of tables, in which the 

 number tabulated is only approx{nlately*c6rVe'ct, is whether it is preferable 

 to apply proportional parts to the true calculated difference of the 

 function, or to the actual tabular difference. Thus in the seven-figure 

 logarithms 



log 66310 = 4-8215790 



log 66311 = 4-8215856 



the tabular difference is 66, while the correct difference as obtained fi-om 

 Vega's ten-figure table is 65494, which, to seven figures, is only 65. The 

 question is, whether it is better to use 65 or 66 for finding the proportional 

 parts. By a comparison of the extreme cases, M. Lefort* has shown that 

 so long as the tabular difference (that is to say, the difference actually 

 found between the numbers as given in the table) is used, the last figure 

 in the interpolated result cannot be in error by more than unity ; while 

 if the true calculated difference, cut down to the nearest figure, be used, 

 the last figure may be in error by more than unity. It follows that the 

 actual difference of the table, and not the mean difference given in some 

 tables (such as Hutton and Callet), should be used for the interpolation. 



Section 9. — Interpolation of Douhle Entry Tables, or Functions of two or more 

 Variables, 



The problem of the interpolation of functions of two variables presents 

 but few diflBculties beyond those of interpolation of functions of a single 

 variable, excepting what i^ dije.tp J;he increased complexity of the process. 

 This renders many of the special artifices practically iinmanageable, 

 although the very fact of the complexity increases the importance of 

 simplifying the actual work. Nevertheless, it is better on the whole, in 

 processes which are not of frequent use, to encounter deliberately a little 

 excess of arithmetical labour, rather than to risk the chances of error from 

 want of analytical simplicity and perspicuity. If any one process should 

 be often wanted, those who need it may be left to invent the special 

 machinery. 



The general theorem of interpolation for double entry is expressed in 

 the two formal identities. 



«,, = (l + A)-(H-S)"«oo 



in which A refers to variation with regard to x, on the supposition that y 

 is constant, and Z refers to variation with regard to y, on the supposition 

 that X is constant. 



In general it is usual to give equal weight to the two variations, that 

 is to say, that if it is proposed to neglect terms of the order p, then all 

 terms of the form A'" o", where in -f- n =2^, are to be discarded, whatever 

 may be the separate values (always supposed positive) of m and n. The 

 terms of the expansion would thus be grouped as 



■Wx,v = «o 

 * See the Proceedings of the Eoyal Sac. of JEdinhnrgli, vol. viii. (1875), p. 610. 



