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



method gives a fair general view of the temperature corresponding to a given time and 

 position, and it can, of course, be employed with equal facility when the boundary 

 temperatures vary in almost any assigned manner, including cases that would be 

 difficult or impossible by the Fourier method. 



3. It frequently happens that the integral can only be determined with reference 

 to the boundary as a whole, as, for example, in the calculation of the electrostatic 

 potential at all points of a region when its value is given over the conducting 

 boundaries. Here the differential equation is of the second order, and the first space 

 rates of its integral are not given on the boundary, so that the step-method is 

 inapplicable. The following 3 contains an account of two methods for solving 

 problems of the type indicated. 



3'0. The Determinate Nature of the Problem. Let there be n body-points and 

 .9 boundary-points in the region considered. Then the differential equation, to be 

 satisfied in the body, is approximately represented at any body-point by an algebraic 

 equation connecting the body value there with the surrounding values. This algebraic 

 equation will be of the first, second, or higher degree, according as the differential 

 equation is of the first, second, or higher degree in the function of position and its 

 differentials. Forming this equation at every body-point, we have a system of 

 n simultaneous integral equations between s + n unknowns. To make the problem 

 determinate, the boundary conditions must therefore supply s independent relations, 

 involving the boundary values. The rules governing the 

 arrangement of these s boundary equations, so as best to TABLE II. 



represent the given infinitesimal boundary conditions, have 

 not yet been elaborated. In certain cases a choice of ways is 

 open, as in the following example : Let the body equation be 

 (3 2 /3^ 2 +3 2 /3?/ 2 ) f = p, where p is a given function of position, 



and the boundary condition df/dn + ^f 0, where is an 



arbitrary function of position on the boundary. Let the body 



values of f be denoted by t//i, // 2 , ..., and the boundary values 



by /3 1; /3 2 , &c. Then in the annexed Table II. we are at liberty to choose between 



two alternative approximations. For we may take as values of f on the boundary 



And as corresponding values of 



So that we have one boundary condition for each /3. Or else we may suppose the 

 corner slightly bevelled, so that, while the above relations still hold for /3 t and /3 4 , 

 we now have at the corner /= \ {\jj 2 +^(/3 2 + (3 s )}, 1 S//'Sn = \ (/3 2 +/3 3 ) t| 2 . Here 

 A+As enters as one variable. Also /3 2 and /B 3 only enter the body equations 

 when combined in the form /3 2 +/3 3 . We no longer seek to determine /S 2 and y8 3 

 separately, but merely their average, which is the value off just outside the corner, 



