﻿550 Mr. W. Ellis Williams on the 



number of tabulated values of %(?*)• This function is of an 

 exponential form except when r is small, and it was found 

 that the usual interpolation f ormulse (Newton and Gauss) did 

 not give sufficiently accurate results, since, being based on 

 Taylor's theorem, they fail when the successive differential 

 coefficients do not diminish fairly rapidly. A very simple 

 interpolation formula may, however, be based on the assump- 

 tion that the functions are exponentials with slowly varying 

 indices. Thus suppose a, b, are two consecutive values of 

 the variable in the table, and f(a),f(b), the corresponding 

 values of the function which is to be integrated, and let a-\-h 

 be the value of the variable in the step a — b, h varying con- 

 tinuously from to (b — a) ; then we may put 



f(a + h)=f(a)e k \ 



k being a different constant for each step in the table. 



. C# ^i\ /(«)*** 



hence 



Ja 



+h) =/(*)-/(") 



and the index k is given b\ 



e> 



, 1 log /w 



b — a & f(a) 



The step (b — a) is usually unity, and hence k is found at 

 once by subtracting the logarithms of the two consecutive 

 terms in the table, and the step of the integral is got by 

 subtracting the two terms and dividing by k. The inter- 

 polation can thus be carried out without any very laborious 

 computation. 



The calculations have been carried out for two values of X, 

 namely X=*09 and \=1. The first represents a very slowly 

 changing velocity and is not very different from the case of 

 steady motion, while the latter value gives the case of very 

 rapidly accelerated motion. The results obtained are given 

 in the accompanying table showing the values of <fi (r) and 

 (f>i(r) for different values of r. The value of ^jr at any point 

 is obtained by substituting in (18). 



In figs. 11 and 1*2 these values of -v/r have been plotted out 

 for X = l and for two values of the velocity, one above and 

 one below the critical value. It will be seen that the change 



