BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 



315 



so that there is still one boundary equation for each boundary variable which we seek 

 to determine separately. We have just considered a blunted angle of 90 degrees. By 

 examining the remaining five angles possible with two co-ordinates, namely, 45, 135, 

 225, 270, 315 degrees, one may convince oneself that with a closed boundary it is 

 always possible to arrange to have just as many boundary equations as boundary 

 unknowns. It is in some cases necessary to suppose the corner slightly bevelled 

 and to replace certain (/3)'s by their averages. The representation of the boundary 

 condition when both /" and df/dn are given at each point will be considered in the 

 theory of the dam. 



When the infinitesimal boundary conditions are such as to make the problem 

 determinate, it will be assumed that we can, and therefore do, represent them by a 

 set of boundary equations equal in number to the boundary unknowns. If any case 

 be discovered in which this is impossible, it will be an exception to the rest of 3. 



3'1. The finite difference problem being thus made determinate the most direct 

 way of finding the integral is to solve the n + s simultaneous algebraic equations for 

 the n body and is boundary values of the integral. To take an example : At 

 one pair of opposite 



edges of a square <f> = 1, I ABLE III. 



at the other pair < = 0. 

 Inside 



0-5 



edge 



of 

 square. 



= o 



0-5 



J 



0-5 



0-5 



0-5 



everywhere. Find < in- 

 side the square. 



Now by symmetry 

 the values of < on the 

 diagonals will be every- 

 where 0'5. In fact, we 

 need only consider ^ of 

 the area of the square ; 

 all the rest follows 

 from it. Taking finite 

 differences, Table III. 

 is drawn up with the 

 given boundary and 

 diagonal numbers in 

 their proper places and 

 a, b, c, d, e, f for the 



unknown body values. Now as the finite difference expression for V 2 ^ given in 1*1 

 has to vanish at all body-points, we have a relation between each of the letters 

 a to f and its four nearest neighbours. 



2 s 2 



centre of 



0-5 

 square. 



* 



edge 

 of square. 



