424 Transactions. — Chemistry and Physics. 



It is to be noticed that all the terms except the parabola 

 have maximum ordinates of about Ol, so that two places of 

 decimals in their factors is ample. The quartic function, the 

 term of which has here practically disappeared, has a maxi- 

 mum of O0625, so that the quantities it represents are never 

 larger than 0-0013 in this interpolation, a quantity which is 

 negligible with respect to the second place of decimals. This 

 is against 0-052,0 in Bessel's method. 



Comparing this with the Bessel's computation, we see 

 that the fifth-degree term vanished in our table, while in 

 the Bessel's its value was estimated, when it vanished; the 

 fourth-degree term becomes negligible in our computation, 

 while in Bessel's it is conspicuously large. The third-degree 

 numbers are the same, 0-025,2. The Bessel's computation 

 illustrates very clearly what was said about Bessel's method 

 effecting an improvement oniy in the odd power terms. 



It is hardly necessary to remark to those who may ex- 

 amine this method of using differences that there is no im- 

 provement on Newton's or Lagrange's methods for second 

 differences, nor on Bessel's for third ; but for fourth, and 

 especially for higher differences still, the extra trouble of 

 forming the table seems worth while, and certainly is worth 

 while if a number of values have to be interpolated. The 

 method of standard terms also possesses an obvious advantage 

 where the problem is to find at what value of x, Y has a given 

 value, since when fourth differences have to be used the 

 approximation in three terms [Y = A + ~Bx + C(x — x 2 )] is 

 more nearly accurate than the corresponding approximation 

 of any of the other methods. 



Note. 

 The standard functions referred to in this paper are : — 



and the list may be provisionally extended by P-multiplica- 

 tion. This list shows how the values of the functions for any 

 value of x can be easily computed, or they may be taken from 

 a table of values similar to those of the binomial coefficients 

 (Newton's method) or of Bessel's coefficients. Such tables, 

 it is confidently believed, will soon be included in many 

 mathematical tables. 



