of Wu'iable Qu<int'du>.f<. 

 fereucino- with the omission of alternate ordinates, 



Now in the figure let A F, B G, C H, D I, E K be any five 

 consecutive ordinates, and suppose that it is proposed to cor- 

 rect the ordinate C H. Then, if the construction shown in the 

 figure be carried out, it is clear that 



CP=AXBG), and CQ=iA^X^F), 



and therefore 



PQ = AXBG)-iA^XAF). 



Thus the correction to be applied to the ordinate H C is ^ 

 (or very nearly \) of P Q. The same process must be applied 

 all along the series for each set of five points. 



Four points are lost out of the series, two at each end. For 

 example, if A, B are the first two points, the rule gives no 

 substituted points on those ordinates. The results obtained 

 from the use of this rule do not seem markedly superior to 

 those given by the empirical method, except where the points 

 lie in a fair curve ; and as the rule is more cumbrous to apply, 

 it does not seem likely to be of much practical value. 



The construction of a fair surface near a number of points. 



The preceding process may be extended to the case where 

 the function involves two independent variables. The observed 

 values may, for the sake of clearness, be considered as con- 

 sisting of a number of ordinates standing on the intersections 

 of the lines of a chess-board, of which two intersecting edges 

 are the axes of x and y. 



Let [j', y~\ indicate the given ordinate which stands on the 



* If the points lie in a fair curve, ^'^yx-\-\^''^yx-2 is very small, a 

 property whicli I have used to give a rule of graphical interpolation on 

 intermediate ordinates (see the 'Messenger of Mathematics,' .January 

 1877). Thus in this case the correction applied is very small. 



