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IX. The Approximate Arithmetical Solution by Finite Differences of Physical 

 Problems involving Differential Equations, tvith an Application to the 

 Stresses in a Masonry Dam. 



By L. F. RICHARDSON, King's College, Cambridge. 



Communicated by Dr. R. T. GLAZEBROOK, F.Ii.S. 



Received (in revised form) November 2, 1909, Read January 13, 1910. 



1. INTRODUCTION. 1 - 0. The object of this paper is to develop methods whereby 

 the differential equations of physics may be applied more freely than hitherto in the 

 approximate form of difference equations to problems concerning irregular bodies. 



Though very different in method, it is in purpose a continuation of a former paper 

 by the author, on a " Freehand Graphic Way of Determining Stream Lines and 

 Equipotentials " ('Phil. Mag.,' February, 1908; also ' Proc. Physical Soc.,' London, 

 vol. xxi.). And all that was there said, as to the need for new methods, may be taken 

 to apply here also. In brief, analytical methods are the foundation of the whole 

 subject, and in practice they are the most accurate when they will work, but in the 

 integration of partial equations, with reference to irregular-shaped boundaries, their 

 field of application is very limited. 



Both for engineering and for many of the less exact sciences, such as biology, there 

 is a demand for rapid methods, easy to be understood and applicable to unusual 

 equations and irregular bodies. If they can be accurate, so much the better ; but 

 1 per cent, would suffice for many purposes. It is hoped that the methods put 

 forward in this paper will help to supply this demand. 



The equations considered in any detail are only a few of the commoner ones 

 occurring in physical mathematics, namely : LAPLACE'S equation V 2 </> = ; the 

 oscillation equations (V 2 +F)< = and (V 4 -& 4 )< = 0; and the equation V 4 ^. = 0. 

 But the methods employed are not limited to these equations. 



The Number of Independent Variables. In the examples treated in the paper this 

 never exceeds two. The extension to three variables is, however, perfectly obvious. 

 One has only to let the third variable be represented by the number of the. page of a 

 book of tracing paper. The operators are extended quite simply, and the same 



VOL. CCX. A 467. 2 R 2 24.5.10 



