422 Transactions. — Chemistry and Physics. 



Converting these functions into standard form, we get the 



the first difference 



VI. 



900 256 



(The fractions are given as products to facilitate computation.) 

 Thus a value of the Lagrange A*, for instance, equal to n, 

 gives rise in the interpolation formula to the terms — 



»(*(P)-*(IV.))- 



From this we see that, although it may be necessary to 

 take the differences out to a high order, it does not follow 

 that the formula necessary will be of high degree. 



From this table we can at once see the improvement that 

 Bessel made in Lagrange's formula, for it is evident that, 

 taking half of the odd differences out in the term depending 

 on the preceding even difference results in the elimination 

 of the fractions which occupy the spaces in lines A 3 , A 5 , &c, 

 column P, lines A 5 , A 7 , &c., in column IV., and so on. 

 Thus Bessel's third difference function is the standard cubic, 

 and the fifth a sum of cubic and quintic, which is an obvious 

 improvement upon Lagrange's method. 



It is not a complete improvement, for it does not alter to 

 any appreciable extent Lagrange's even difference terms; and 

 even with respect to the odd differences it includes a cubic 

 term in the fifth difference function, instead of making it a 

 pure standard quintic." 



If we compute by means of the table given above, it is 

 clear that we are able to effect completely what the Bessel 

 method effected partially, in giving the terms of the formula 

 nearly the best possible form — that of the standard functions. 



A concrete example of a problem treated by Lagrange's 

 and Bessel's methods and by that of the standard functions 

 will serve to show the advantages of the latter method. 



Loomis's " Practical Astronomy," 1894, p. 207, gives a 

 convenient example, that of getting the moon's E.A. from a 

 twelve-hour table, for eight hours {i.e., x = §-). Two places 

 of decimals are obviously enough, but we use three for the 

 purposes of illustration. 



* We here assume that the standard functions are the best practic- 

 able formulae for convergence, concerning which vide the paper quoted. 



