340 MR. L. F. EICHAEDSON: APPEOXIMATE ARITHMETICAL SOLUTION 



especially as the arithmetic was somewhat faulty. Let us start with the table as left 

 by them and call it </>i. 



Next the processes recommended in 3'2'1 were adhered to. The table being large 

 it was thought best to remove PI and P 2 from fa (j) u = SA t P A by making guesses at 

 the form of PI and P 2 , calculating their coefficients Aj and A 3 by the Fourier process 

 and then subtracting A^ + AaPa from <i <. Of course P A is defined so that 

 SIP* 2 = 1, where S denotes a summation over the body-points, whereas the guess is 

 an approximation to B^P*, where B is an unknown constant. For $/ = ^ 4 , and for 

 zero boundary values of the (P)'s the general equations of the appendix give 



1=1, x, 8 = V/T = S [> ' (B t P 4 )]*/S [B A PJ. 



Now ^ 4 <, = SX/A.P*. Therefore S [P* . ^ 4 <fc] = X/A*. Therefore 

 A,P, = P, . S [P, . F ty] x S [B A P,] 2 /S [ F 2 (B,P,)] 3 , = (B,P A ) . S [B A P t . ^ ty]/S [ ^ 4 (B*P,)]'. 



In the right-hand side P* only appears in the combination B A P A , so when this is 

 known A/.P* can be calculated at each point. A rough check on the accuracy of the 

 guesses at B^ and B 2 P 2 may be obtained by comparing the values of A/ and X/ 

 obtained from them, namely, 0'18 and 2'3, with the corresponding quantities in 

 infinitesimals for simpler geometrical figures of the same area. The area of this 

 smaller dam table is 113 square units. And for a rectangle f as broad as long the 

 formula of 3'1 gives \* = 0'15, while for a circular plate clamped at the edges the 

 principal vibration with a single circular node lias X 2 = 1'2 (PtAYLEiGH, ' Theory of 



Sound,' 22lA). It is possible that what 

 has been called X/ above was really X 3 2 . 



Finally the table was given four approxi- 

 mations of the type x+i = X-.-"' 1 ^^- 

 where a was 10, 30, 50, and 64 in turn. 

 These numbers were chosen because they 

 gave a good " curve of ratio of final to 

 initial amplitude." The result is shown in 

 s s >y Table X. From it the stresses can easily 



be calculated. The values of ^ 4 ^ which 



should vanish are given there also. 



4'2'2'L -Interaction of Body and Sur- 

 face Equations. Bevelling of Re-entrant 



xxx Angle at Front. It has been shown in 



4'1'9 above that the given stresses on 



the upper surface determine a double layer of values of x covering this surface. 

 At almost all points of the double boundary layer it is impossible to evaluate ^ 4 x 

 because one or more of the values of x involved is lacking, so that x over ^ ne 

 surface must be determined by the surface condition only. The only exceptions are 

 near a re-entrant angle. For example, at the points where ^ = a or 6 in. the 



