10 



Mr. Gr. H. Darwin on Fallible Measi 



ires 



point X, y, let E, A be the operations of writing x-\-l for og, 

 and of differencing with respect to x ; and let E^ D be the like 

 operations with respect to y. Let (/> represent a single smooth- 

 ing operation on any series of ordinates which are in a plane 

 parallel to x ; and -^ the same with respect to y. 



Now apply a smoothing operation </> to all the points pa- 

 rallel to X, and then apply the operation i/r to all these new 

 points. The order in which these operations are performed 

 is immaterial : and the result is 



<i>^lx,y'] = 



E^ + E-^ E' + E- 





2 2 



This, interpreted geometrically, means that we are to erect in 

 the middle of each square an ordinate which is the mean of 

 the four surrounding ordinates. Again, 



+ [•^-1. 3/-1] + 2([^ + 1; ,y] + [^^-1, y'] + l^, y + l] 

 + ['^',y-l]) + 4[^,y]}. 



If the figure represents any four squares 

 of the chess-board (in which observe that 

 nine ordinates stand on the intersections), 

 the rule given by a double operation is to 

 substitute for the ordinate at K, 



Y^ of sum of ordinates at A, B, C, D 



-I- 1^ of sum of ordinates at E, F, G, H 



+ \ of ordinate at K. 



It is clear that the operations of smoothing parallel to the 

 two axes are quite independent, and that there is no necessity 

 to smooth the same number of times in each direction. The 

 symbolical w^ay of writing the operation makes it perfectly 

 easy to construct any desired modification of this formula, 

 where </> and -x/r are each performed any number of times. 



The process may also be extended with equal justice to the 

 case where there are three independent variables, although 

 that case no longer admits of geometrical interpretation. 



Let ^ be a third variable; then, if the former notation be 

 extended, and if only a type of each form of term be written 

 down preceded by the sign of^summation,it will be found that 





A 



E 



B 







K 



F 





H 







D 





G 



C 



<^'tY[^,i/,<]=A{2[,t-±l,y±l,<±l]+22[.,.,y: 



1,': 



