BY FINITE DIFFEEENCES OF PHYSICAL PROBLEMS, ETC. 319 



commoner possible forms of 2)' will be found in the table on p. 354. As well as the 

 equation 2>'< = 0, the equation %'<$> = (a given function of the co-ordinates), and 

 the equation (<)' \ 2 ) <f> = may be treated by these successive-approximation- 

 methods. They will be discussed in order. 



3'2'1. The Equation >'( = 0. The approximation process proceeds as follows. 

 Let <f) u be the correct finite-difference integral. Let < t be a function (that is a table 

 of numbers) satisfying the correct boundary-conditions, but arbitrary as to its body 

 values. Next calculate the body-values of (j) 2 by means of 



k = ^-ar 1 s>'& .......... (i) 



where as.^ is a number to be fixed ; and fill in such boundary- values of ^ as will satisfy 

 the same boundary-conditions as <. The succeeding steps are each of the form 



<+! = fa-**' 1 $'< .......... (2) 



for the body values, and by choosing the boundary values </>,,, +1 is made to satisfy the 

 correct boundary condition. These are matters of simple arithmetic. It will be 

 shown that by the judicious choice of a,, a 2 , ..., u t it is possible to make < (+1 nearer to 

 <j> u than (^ was. For since X>' is linear and >'< u = we have from (2) 



Now it is shown in the Appendix that (f> m (j> u may be expanded in a series of integrals of 



(5/-V) P 4 = (4). 



Put <i -(/>,,= SA^P* (5). 



Then by (4) & (<t>\ <!>,,)= +SA A .\ A 2 P A . 



And therefore by (3) < 2 -< = SA A (l - ) P k . (G). 



i/ 



Proceeding in the same manner after t operations we arrive at 



A measure of the deviation of two functions from one another which is used in the 

 theory of Least Squares is the sum of the weighted squares of their differences. On 

 the same principle let us measure the error of <f) t+l by 



E t+1 '=S(<k +1 - u ) 2 xI (8), 



where S stands for a summation over the body points and I is a certain one-signed 



