124 



E. L. De Forest, — Un symmetrical Lun^ of Error 



side of ^l^^. {Analyst, ix, 135.) If now u is a function of both an 

 abscissa aAx and an ordinate hAy, so that the terms form a double 

 series or rectangular table, thus, 



«_i 



'-2.-1' "-!■ 



"no? "^"10' 



1^ (4) 



the formula for any desired term ?/„.,, is 

 a , a2 . «.(a3_i2) 



-^,.0 + 



aft 



a^b . a(a^-]^)b 



1 0-1 ].l *"■ 1.2.1 --1 1.2.3.1 



a2j2 



Txr2 



52 , «^~ , 

 1.2 '••-1.1.2 

 4- 



-J3.1 + 



+ ^^— Jo:a+ \^., /^ -^8-2+ • • • 



(5) 



1.2.3.1.2 



The coefficients of the differences in the upper row are the same as 

 those in (3), and those in the left hand column likewise, only substi- 

 tuting b for a, Avhile the coefficient for any other difference is the 

 product of the corresponding ones in the upper row and the left hand 

 column. For example, the coefficient of Z/3 3 is the product of those 

 of Jg.Q and ^o-2- '^^^^ reason is, that the values of « in (4) are sup- 

 posed to represent ordinates to an algebraic surface, and conse- 

 quently any one row or column will i-epresent equidistant ordinates 

 to an algebraic curve. Denoting any term in the middle column by 



U(i 4, its value by (3) is 



b . b 



-J„ o+etc. 



(6) 



and the value of any term ii^j, in the same row with i<o j is found by 

 applying (3) to each term in the second member of (6), giving the 

 expression (5). Any difference z/„,„ is the result of differencing ni 

 times in the a direction and 71 times in the b direction, so as to keep 

 u^ Q always in the middle. If the difference in either the a or the b 

 direction is of an odd order, we must take half the sum of the two 

 nearest corresponding differences on either side of Wq.q; while if the 

 difference in both directions is of an odd order, we take the mean of 

 four differences nearest to «^ g. For example, 



■M -2"i.o +«"o.o -6?<-i.o 



+ 2y_ 



^2-1 "'"1-1 



■6?/., 



+ «3._i— 3*^,._j +3Wo.. 



-« 1.1 



(7) 



