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



for every pair kl. This is the condition that the body equations can be derived 

 from a single function as in (2). The numbers i,, i a , ... i n may be regarded as the 

 body values of a function of position. It will be denoted by I and called the 

 " unifying factor," because it allows all the body equations to be expressed in terms 

 of a single function V. 



For the sake of a certain transformation, which will be introduced later, V must be 

 quadratic and homogeneous, and therefore both body and boundary equations must 

 be linear and homogeneous. 



Thus let 



........ (5). 



+<Ws+ ...e*,& 



And lot the boundary equations be 



........ (6), 



where the ('')', ('')' >s > and (/)' s :ir e constants, most of which will commonly be zero. 

 In many physical problems we have the integral <f>,< given on the boundary. This 

 corresponds to a term j^ > independent of the (t|)'s, in each of the equations represented 

 by (G). However, it is not < which we wish to expand in the form SA A P A , but the 

 difference (<,<) between the integral and an arbitrary function satisfying the same 

 boundary conditions. In <f> m <j> u the f^ terms cancel. Thus we are only here concerned 

 with functions having boundary equations which are linear and homogeneous in the (i|)'s. 

 Next, equation (4) leads to the following relations between the coefficients of (5) 

 and (6) : 



_ .1 = S _ J ' = t 



'$,e kj f] l ....... (7) 



for every pair k and /, making at most fyi (n l) equations. Now it is customary in 

 physical mathematics to treat body and boundary conditions separately. Let us 

 adopt the same course here and treat only the case in which (7) splits into two parts, 

 namely, 



to be satisfied within the body region, and 



]=> 



*i 2 eyfjt = i k S eqfp (9) 



j ' = i j = i 



to be satisfied on the boundary. 



In the centre of a sufficiently large table (7) reduces to (8) even in the most general 

 case, for there the (e)'s and (/)'s vanish. 



Stated in words, i,c tt = i k c kl means that the body equations must be able to be 

 brought into forms such that if each pair of points in the table be taken in turn, then 

 the integral <f> at the first point of the pair, enters into the body equation >'< at the 



