310 MR. L. F. RICHARDSON: APPROXIMATE ARITHMETICAL SOLUTION 



which there is a value of <f> which enters into the system of body-equations by way of 

 the boundary-conditions will be called a boundary-point. 



The values of any function of position at these two classes of- points will be 

 distinguished as body values and boundary values, or synonymously as body-numbers 

 and boundary -numbers. 



Problems are divided into two main classes according as the integral can or cannot 

 be stepped out from a part of the boundary. They are discussed in 2 and 3 

 respectively. 



1'2. Errors due to Finite Differences. Having solved an equation using the 

 simple expressions of I'l for the differential coefficients, it remains to enquire how 

 much in error the integral may be. A rule of apparently universal application is to 

 take smaller co-ordinate differences and repeat the integration ; and, if necessary, 

 extrapolate in the manner explained below. 



It is known* that when central differences are used, the expansions of the differential 

 coefficients of a function in terms of its differences contain only alternate powers of 

 the co-ordinate difference h. The same is true for partial differential coefficients and 

 for products of differential coefficients. Consequently the error of the representation 

 of any differential expression by central differences is of the form A 2 F 2 (x, y, z) 

 -f/t'F, (x, y, z}+ terms in higher powers of h 2 , where F 2 , F 4 , &c., are independent of A.t 



Next, as to the error of the finite-difference-integral <. This is the infinitesimal 

 integral of a differential equation having the error h*F i (x,y,z)+h f F t (x i y t z) + i &e. 

 Let P be the integral of the correct differential equation. Then, if we write 



< ( x > V> z ) = V ( x , y> z ) + m $i ( x , y> z ) + m ^ K > 2/> z ) + terms 



in higher powers of in. it follows that a differential expression of any order and degree 

 for (f> differs from the corresponding one for P by 



m x (a function of the differential coefficients of 9 and of t/j)+ terms in m a , m 3 , &c., 



provided only that m is independent of the co-ordinates. Now. identifying m with A 2 , 

 it follows that : the errors of the integral and of any differential expressions derived 

 from it, due to using the simple central differences of I'l instead of differential 

 coefficients, are of the form 



h% (x, y, z) + h% (x, y,z) + h% (x, y,z) + ,&c. 



Consequently, if the equation be integrated for several different values of h, 

 extrapolation on the supposition that the error is of this form will give numbers very 

 close to the infinitesimal integral. When h is small enough the error is simply 



* W. F. SHEPPARD, "Central-Difference Formulas," 'Proc. Lond. Math. Soc.,' vol. xxxi. (1899). 



[t Note culded January 21, 1910. It is assumed that the co-ordinate axes in the tables which are 

 compared are parallel, for the error at a fixed point and for a fixed value of h may depend on the direction 

 of the axes.] 



