316 ME. L. F. EICHAKDSON: APPKOXIMATE ARITHMETICAL SOLUTION 



The solution of these six simultaneous equations was accomplished in an hour and 

 gave the following results : 



The numbers for infinitesimal differences were obtained from 



I 4 ,_, / ., >^ ? 1 , IllTT i 



rf> = - 2 ( 1) a sech cos mx cosh mz, 



TT , odd m 2 



the separate terms of which satisfy V 2 <f> = at all points, and (f> when a 1 = ^TT, 

 and by their addition make <f) = 1 when z = ^TT. Adding up the series at these six 

 points took 3 hours. It is seen that the greatest error is 1'4 per cent, of the range of 

 potential between the side and diagonal. 



Further, if we take co-ordinate differences of twice this size, leaving only one 

 unknown in the same position in the square as e here occupies, its value is easily found 

 to be 0'6250. Extrapolating as in 1'2 we find for infinitesimal differences at this 

 point </> = 0-G324 + ^(0'6324 0-6250) = 0'G348, and this is only -j^th per cent, in 

 error. To correct the other values we should have to halve the co-ordinate difference 

 instead of doubling it, and this would require much more work. 



As a second example of the use of simultaneous integral equations, let us take the 

 determination of the gravest period of vibration of a thin square plate, with edges 

 clamped, in a plane. It is known (LovE's 'Elasticity,' ed. 1906, p. 469) that the 

 displacement normal to the plate is of the form W cos (pt + e), and W satisfies the 



/ r) 4 3' 1 S 4 \ 



equation ( +2 , . + I W = V^W = c 4 W, where c 4 is given as a function of jp, 



the elastic constants, the thickness and the density. Now let us form a table such 

 as (IV.) to represent W. In this table W is measured from the plane of the clamped 

 boundary. The differential equation, when turned into finite differences, becomes a set 

 of simultaneous equations, connecting each in turn of the unknown body values q, r, s, t 

 with its twelve nearest neighbours. As the boundary numbers are all zero there are 

 no constant terms in these equations, and they are only consistent when the 

 determinant of the coefficients of q, r, s, t vanishes. There are a number of values 

 of c 4 which cause the determinant to vanish, and of these the smallest is that 



