BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 325 



Again, from the size and distribution of 2)'( combined with a knowledge of various 

 integrals of 1)'</> = f(x, y), a rough estimate of the errors in <j> can frequently be made. 

 Again for certain equations* the method of contour integration applied to a circle 

 affords a check on the value of (f> at its centre. This method is very rapid, and it is 

 particularly advantageous when applied to a circle enclosing many body values, for 

 then a repetition of the approximation process would be correspondingly tedious. 



3 '2 - 5. Routine of Approximation. Time and Cost. To anyone setting out on a 

 problem I offer the following experience as a guide in forming estimates : It was 

 found convenient to enter certain stages on a table with 



O flp 



large squares, each divided into compartments. Thus for 



^j +2 + ^~T = 0. one of the squares is shown in 



3 4 Kx^&if % 4 



the annexed table. All the quantities in it refer to the 



. fc> , 2 i f& 4 , 



central point of the square. The intermediate stages are 

 done on rough paper and thrown away. So far I have 



</>3 &C. 



n 

 paid piece rates for the operation %/+% y a of about T^ 



pence per co-ordinate point, n being the number of digits. The chief trouble to the 

 computers has been the intermixture of plus and minus signs. As to the rate of 

 working, one of the quickest boys averaged 2,000 operations 'S I a +'S y 2 per week, for 

 numbers of three digits, those done wrong being discounted. 



3 '3. Relative Merits of Simultaneous Equations and of Successive Approxi- 

 mation. The method of simultaneous equations may be applied to differential 

 equations of any order and degree. It gives results which are exact for finite 

 differences. It is necessary in discussions as to the existence and properties of the 

 integrals of difference equations. But for actually calculating the integrals the 

 labour becomes very great as the number of unknowns increases, and is of a sort 

 which a clerk will not easily do. Large numbers of digits have to be dealt with, and 

 a single mistake generally throws the result altogether out. 



The successive approximation methods of 3 '2 have only been applied to a limited 

 class of linear equations. The results are not exact even for finite differences. But 

 the bulk of the work can be done by clerks who need not understand algebra or 

 calculus. Small and infrequent mistakes, or taking only a small number of digits, do 

 not prevent one arriving at a fairly correct result. Nevertheless, it has been found 

 best to have everything worked in duplicate. 



The method of successive approximation to the surface z = <f> u =f(x,y) reminds 

 one of the manufacture of plane metallic surfaces. The initial form of the surface is 

 arbitrary in both cases. The essential things in both cases are a method of testing 

 the work at any stage, a tool with which to alter the surface and judgment in using 

 it. Methods of testing the arithmetic have been described in 3 '2 '4 above. Our 



* These include V 2 < = 0, V^ = 0. See a paper by BOGGIO, ' Jahrb. Fortschritte Math.,' 1900, p. 740. 



