BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 



317 



belonging to the gravest mode of vibration. Let W l =f(x, y) be the appropriate 

 integral for a square of unit side. Then W L =f(x(L, 7//L) will bear a similar relation 



to a square of side L and to a new constant C L *. Then C L 4 = * L = ? - 1 = ^ 



TABLE IV. 



V 



L 4 W, L 4 



so that L 4 C L 4 is independent of the length of side of the square. It will be interesting 

 to notice how this constant con- 

 verges towards a limit as the 

 number of co-ordinate differences 

 in the side of the square is 

 increased. / 



The configuration chosen was 

 one in which the sides of the 

 square are at 45 degrees to the 

 rows and columns of the table. 

 This gives a sharper boundary 

 than the parallel arrangement. 

 The symmetry of the gravest mode 

 rediices the number of unknowns. 

 As well as the arrangement in 

 Table IV., two smaller ones were 

 also considered, namely, those 

 formed by cutting off in turn 

 its first and second outer layers. 

 C 4 was calculated in each case 

 from the determinant, by approximation where necessary. 



Collected results : 









Side of square = L. 



IK/2 



C L 4 for the gnivest mode. 



20-0000 

 6 3058 

 1-88843 



405 0000* 

 9S5 2iS 

 11 33 '53 



We may attempt a closer approximation by assuming that the error in C L 4 is 

 inversely as the square of the number of co-ordinate differences in the side of the 

 square, in accordance with 1'2. From the 1st and 2nd values extrapolation gives 

 L 4 C L 4 = 1311-69. And from the 2nd and 3rd L 4 C L 4 = 1287'96. Or, if we assume 

 that the error is of the form e 2 h 2 +eji* and extrapolate from L 4 C L 4 for the three values 

 of h, we find L 4 C L 4 = 1282'62. The way in which the values of L 4 C L 4 converge 



* Revised figures, March, 1910. 



