BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 345 



upper boundary infinitesimally correct as regards x, when the value of x was found 

 by interpolation* (%xfi ( l appeared to be near enough already). And this correction 

 was used to bring the lower boundary of the fine-difference table into harmony with its 

 upper boundary. Its greatest value in the region of the fine-difference table was 1'5. 

 Secondly, a principal mode of vibration, having maximum value 0'5 and X 3 , roughly 

 1'5, was left in the top of the dam not completely removed by the approximation 

 with a = 2. It was removed by guesswork satisfactorily. Thirdly, a number of 

 slips in the arithmetic had to be corrected. The result of these processes is shown 

 in Table XII. 



4 "2 - 3 '4. Errors Due to Incomplete Approximation. As a justification of the 

 body values of Table XII., I propose to consider the final distribution of ^i 4 x 

 belonging to them, and not the process by which they were obtained, for owing 

 to errors and experiments this process was long and complicated. Let us imagine 

 the distribution of j^i'x given in Table XII., to be expanded in the series 

 T6^i 4 X = AjAiTj + A 2 A/P;) +... + ... + A n X,, 2 P, where P A is an integral of 

 (lWi 4 - x /. 2 )I\- = > { 1 lia8 P/i = aud SiyS (normal) = on the boundary. Then 

 the error in x is A?! + A 2 P 2 + ... + A B P M . An inspection of ^i'x shows that it changes 

 sign many times in a small area. Hence it obviously contains the principal modes of 

 vibration for which A a is large. The greatest value, namely, A,,", is about G4, so for 

 this the error in x will be -^ of yg- ^ /x- The greatest value of yg- ^ ^x is seen to be 

 1'81. corresponding to an error of 0'03 in x- If this had been distributed all over 

 with alternate + and - - signs the corresponding error in the stresses would be 

 (4 x 0'03)//i a = 16 x 0'03 = 0'48. This is appreciable, but not serious. The mean 

 value of ye [^ i 4 x> formed by squaring, adding, and taking the square root, would be 

 much smaller than 1'81, and the error in the stresses due to the presence of the 

 higher modes of vibration P re , P B _i, P n - 2 , &c., will be correspondingly less than 0'48. 



Next, as to the gravest, mode Pj : it is possible that PI may be prominent in the 

 error in x, and yet A^Pj obscured in ^ /x by the presence of A B 2 P B , since X, 2 is a small 

 fraction of A/. We can estimate it very roughly by means of the theorem that at 

 the centre of a circle an arbitrary function f differs from an integral x of V*x = 0, 

 which coincides with it, as to value and normal space rate, everywhere on the 

 circumference, by the integral over the interior of the circle of the product 



where a is the radius of the circle. The integration has been effected with sufficient 

 accuracy by drawing contours of B on tracing paper and laying it over Table XII., 

 and then adding up the values of ^ ^x situated between each pair of contours. 



r The internal distribution .of this correction was calculated by contour integration applied to freehand 

 graphs in a way which the author hopes to publish shortly. 

 VOL. CCX. A. 2 Y 



