BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 333 



We proceed to show that the natural way of turning a corner in finite differences 

 is consistent with (11) and (12). Take, for example, the corner in Table VII. 

 representing the tail rising vertically from the base. On the horizontal surface 



*\2 ^2 



z = 0, the equations ^ = 0, - ^- = are taken along the line z midway 



between the two boundary layers. When we come to the corner we may either 



3 2 v 'S 2 v 



continue this process and so determine, b = 5, or else we may take 15-3 +/**/* - = 



Ot// Gtl' OZ 



at x = +|- ; z = 0. 



Either process leads to x = 3 at x = ^, z = , but one fixes b while the other 

 leaves it arbitrary. Again we get x ~ 1 a ^ x ~ ~~ih 2= ~l> either by taking 

 3 2 x/3z 2 +3 2 x/32-3z = at a; = 0, 2 = -1, or else by taking b = 5 and 3 L 'x/3r = 0, 

 and 3 2 x/3.x32 = 0, as holding at the middle of the two columns on x = 0. We 

 may expect that when b is determined by the boundary equations a sharp corner is 

 represented. When l> is not so determined, a corner bevelled by the line x z = ^. 

 In the absence of the foregoing analysis 



these ways of turning the corner would lABLE V 1 



have seemed tempting but risky. But we 

 shall see that they really do correspond to 

 the analytical way. For, on the horizontal 

 bed, we have (3x/3.s-) u = 1, (3x/3<z) u = -2, 

 and the angle turned through is +^TT. 

 Therefore, from (11) and (12) 3x/3.s' and 

 9x/9# on the tail above the corner should 

 have the values 2 and 1 respectively. 

 And these are exactly what we find in the 

 finite difference table for the first differences. 



Again, suppose the tail rises at 45 degrees to the base. Starting along the 

 horizontal the value x = 3 at .c = 1, z = ^ may be obtained by assuming (14) to 

 hold at the origin, that is to say, at the corner. 



The other numbers on the sloping surface were obtained by using (13) and (14) 



alternately, gpz being left out. Now we have (^M = 1, (-^j = 2, a = 77- ; so 

 that equations (11) and (12) give us 0707 and 2'121, respectively, for 



= and -% on the slope iust above the corner. And these are identical with the 

 3s dq 



first differences of the table when Ss and 8q are given their proper values of ^/2 and 

 l/y/2, respectively. 



4 - ] '12. A Possible Experimental Solution. Now that we have shown how to 

 integrate the boundary conditions, the analogy with thin plates will help us. For it 

 is known (LOVE'S ' Elasticity,' 1906, 313) that for a thin weightless plate, originally 



