BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 



311 



proportional to h a . Peculiarities present themselves on the boundary, but it is easy 

 to see that errors will be of the form h a f a +h'f 4 + , &c., provided that in passing from 

 one table to another each part is either kept infinitesimally correct, or else is worked 

 by differences whose size in the one table bears a constant ratio to that in the other. 



An extrapolation can only be made where the tabular points of the several tables 

 coincide with one another. It is conceivable that in the future some method 

 will be found of defining a continuous function in terms of the discrete body and 

 boundary values, so that this continuous function shall have an error of the form 

 h a f a (x ) y,z)+h i fi(x ) y i z)+, &c., everywhere. Extrapolation would then be possible 

 everywhere. 



An excellent illustration is afforded by Lord RAYLEIGH'S account of the vibration of 

 a stretched string of beads (' Sound,' vol. I., 121). He gives the frequency of the 

 fundamental for the same mass per unit length concentrated in various numbers 

 of beads. This is reproduced below in the table. The co-ordinate difference h is 

 inversely as one plus the number of beads, not counting beads at the fixed ends. 



The degree of constancy of the last line shows that if we found the frequency for 

 one bead and for three, then extrapolation, on the assumption that the error is 

 proportional to h 2 , would give us the frequency for the continuous string to about one 

 part in 1000 ; which is as near as we could get by twenty beads and no extrapolation. 

 While extrapolation from the exact solutions for four beads and for nine would leave 

 an error of only one in 50,000. Other examples of extrapolation will be found in 3'1. 



2. Procedure when the Conditions allow the Integral to be Marched out from a 

 Part of the Boundary. 



2 - 0. Histmical. Step-by-step arithmetical methods of solving ordinary difference equations have long 

 been employed for the calculation of interest and annuities. Recently their application to differential 

 equations has been very greatly improved by the introduction of rules allied to those for approximate 

 quadrature. The papers referred to are : 



RUNGE, " tiber die numerische Auflosung von Differentialgleichungen," ' Math. Ami.,' Bd. 46. 

 Leipzig, 1895. 



