326 ME. L. F. EICHAKDSON : APPEOXIMATE AEITHMETICAL SOLUTION 



tools (f> m +i = ^mam" 1 ^'*^ and the Fourier method of removing principal modes of 

 vibration may be compared to automatic grinding machines. The use of the hand- 

 scraping tool corresponds to the adjustment of the numbers according to the 

 judgment of the operator. This is always permissible. 



3 - 4. The treatment of infinity may be illustrated by the following hypothetical 

 example : Suppose we were to set out to determine the field due to a rotating mass 

 of gravitating fluid of known shape, the gravitational potential would have to vanish 

 at infinity. But to be determinable by the methods of 3 the potential would have 

 to be given over some boundary which could be included in the sheet of paper upon 

 which its values were written. We might, for example, assume the potential at the 

 edge of the paper equal to that due to the given mass of the liquid concentrated at 

 its given centre of gravity, and find the figure of equilibrium on this hypothesis. 

 (See treatment of base of dam in 4.) 



4. The problem of the determination of the stresses in a masonry dam has been 

 chosen as a final example for two reasons : (l) Its practical interest arising out of the 

 great expense and productiveness of these structures and the destruction of life and 

 property should they burst ; (2) Its simplicity in that we have to determine a single 

 quantity ^ as a function of two co-ordinates only. 



The methods developed are suitable for finding the stresses, not in dams only, but 

 in a prism or cylinder of any section, acted upon by any distribution of surface stress 

 which is normal to the axis of the prism, when the surface stress is given, provided 

 that the shifts parallel to the axis are zero. 4'1 is devoted to theorems relating to 

 any shape of contour. In 4 '2 the special contour of the dam is introduced. 



The discussion that follows is founded upon two papers in the Drapers' Company 

 Research Memoirs (Dulau and (Jo.) : 



(1) "On some Disregarded Points in the Stability of Masonry Dams," by L. W. ATCHERLEY and 



KARL PEARSON, F.K.S., 1904. 



(2) "An Experimental Study of the Stresses in Masonry Dams," by KARL PEARSON, F.R.S., and 



A. F. CAMPBELL POLLARD, assisted by C. W. WHEEN and L. F. RICHARDSON, 1907. 



I also owe some ideas, e.g., the use of equipollent loads in the base, to suggestions 

 thrown out by Prof. PEARSON in the course of conversation. In these papers the dam 

 is regarded as a prisnaatic-shaped body of indefinite length, so that the problem may 

 be discussed in terms of two co-ordinates x and z lying in the vertical cross-section 

 of the prism. The sluices are ignored a serious omission. 



4 - ri. Prof. PEARSON and his collaborators lay much emphasis upon our ignorance 

 as to the real conditions at the base of the dam and the consequent inapplicability of 

 analysis based upon special assumptions at the base, such as linear, parabolic, or 

 quartic distribution of shear. Admitting this ignorance, there seems to me one 

 assumption more reasonable than any of the others, namely, that the dam may be 



