BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 341 



Table XI. sufficient surrounding numbers exist to determine ^ 4 x completely. 

 We might therefore determine a and b at these points by the body equation. 

 But then we should be left with a boundary strip indicating a considerable 

 amount of bevelling. Or we may go to the other extreme, and determine ^ in 

 the chequer marked c by assuming the surface equation S^/'&e'Sz to hold at the 

 angle. I take it the boundary strip would then represent a perfectly sharp right 

 angle. On these grounds I suppose that the case actually studied, in which a and 6 

 are determined by the surface equations and c by the body equation, represents an 

 angle with the slight amount of bevelling caused by joining the points x = 0, z= ^, 

 and x = \, z = 0. This was not realized till about the 10th approximation, when 

 Mr. BORCHARD pointed out some inconsistency. It was corrected along with other 

 errors' in the boundary conditions, and all final numbers refer to the bevelled angle as 

 just stated. 



4 - 2 '2 '2. Error due to a Point-Force having been substituted for the Actual 

 Distribution. We see from Table X. that the actual distribution of stress differs from 

 the assumed point-force in being spread out well over the base. We may form some 

 estimate of the order of the error involved by comparing the stresses over the lower 

 boundary due to a vertical point-force with the same due to a statically equivalent 

 pressure spread uniformly over the base. MICHELL'S stress function ^ 3 =(r 2 <j) r' 2 <j)')/2Tr 

 enables us to do this. (See equations (22), (23), (24).) For when the point P on the 

 lower boundary is vertically below the centre of the stressed surface, the total force is 

 proportional to the stressed area, which is equal to 2r sin ^a. So that if the total 

 force is to be constantly unity as the area alters, the stress function must be equal to 



. _ _ - y_ g v /23) and (24) the principal stresses at P are 



\ 1 -- . + terms in a 4 , &c. I , -- -- 1- terms in a 4 . &c. I . 

 ir[ 4.3! /' rir\3\ J 



2r sin 



In the limit when the force is at a point we have a = 0, and these reduce to 2/rTr 

 and zero. In the case of a point on the lower boundary, eight units below the level of 

 the rock surface, the base of the dam subtends an angle of about 4J/8 radius. Then 

 a a /4 . 3 ! = 0'013 : that is to say, the errors in the stresses at the lower boundary, due 

 to substituting a point-force for this statically equivalent pressure, spread uniformly 

 over the whole width of the base, are about 2 per cent, of the greater of the two 

 principal stresses at the .lower boundary. This uniform spreading is not, of course, 

 exactly what has happened in the approximation process, but it is sufficiently similar 

 for the question at issue (see Table X.). The correct lake stresses and the correct 

 upper boundary will tend to swamp this 2 per cent., which is therefore quite negligible. 



4 :2 '2 '3. Errors due to Incomplete Approximation. The values of ^ 4 ^ given in 

 Table X. for the last approximation look as though they consisted chiefly of a principal 

 mode of vibration which had a single nodal line sloping parallel to the flank of the 

 dam across the middle of the table. For this mode of vibration X 2 will be considerably 



