THE VALUES OF -?- IV'V*. 271 



WIT J 



and replacing AJ, A 3 , etc., by the second equivalents from (26) we have finally — 



V, = V + n(A + '^ 1 An+^ ) XA 3 +^ 2 A')+ " ( ' t8 - 1 5 f'- 4 > iA» + ^ 3 A«) 



+ / ^^lX^_4)( ? «-9) w , + »-4 A8)+eto (30) 



Either of these formulae, which converge rapidly, may be used for interpolating 

 terms in a series of values already found, especially if we form tables of the values of 

 each term for the various coefficients of A 2 , A 3 , A 4 , etc. Thus, to insert values at 

 intervals of 0"1 between V and V 10 , we have — 



Y 1 =V +^--045.A 2 - -0165. A 3 + 007 8375.A 4 + -003 291 75.A 5 --001 591 0125.A e -etc. ^ 



V 2 =V 1 +^-035.A 2 --0155.A 3 +006 5625.A 4 +003 044 25.A 5 -001365 7875-A 6 -etc. 



V 3 =V 2 +^--025.A 2 -0135.A 3 +'0049375.A 4 + -002 559 25.A 5 --001046 06 25.A 6 -etc. 



V 4 =V 3 + -^--015.A 2 --0105.A 3 + -003 0625.A 4 +001856 75.A 5 --000 656 3375.A 6 -etc. 



M31) 



10 



V 5 =V 4 + y 6 --005.A 2 --0065.A 3 +0010375.A 4 +-000 966 75.A 5 --000 222 9125.A 6 -etc. 

 V 6 =V 5 + A + -005.A 2 --0015.A 3 -0010375.A 4 --000 070 75.A 5 +-000 222 9125.A 6 +etc. 

 V 7 = V 6 + -^ + 015.A 2 + -0045.A 3 - -003 0625.A*- 001 205 75.A 5 + '000 656 3375.A 6 +etc. 

 V 8 = V 7 + -^ + -025.A 2 + -0115.A 3 - -004 9375.A 4 - 002 378 25.A 5 +-001 046 0625.A e +etc. 

 V. =V 8 +4 + -035.A 2 +'0195.A 3 --006 5625.A 4 --003 518 25.A 5 +-001365 7875.A c +etc. 

 V 10 = V, r h^ + -045.A 2 + -0285.A 3 --007 8375.A 4 --004 545 75.A 5 + -0015910125.A 6 +etc. 



If the interval n be — , we have — 

 5 



V 1 = V + 0-2A - -08. A 2 - 032.A 3 + 0144. A 4 + 006336.A 5 - -0029568. A° - etc. ] 

 V 2 =V 1 +0-2A--04.A 2 --024.A 3 +-008. A 4 + -004416 A 5 - -00l7024.A 6 -etc. | 

 V 3 = V 2 +0-2A -008. A 3 +-000896.A 5 -etc. J> (32) 



V 4 = V 3 +0-2A+ 04.A 2 + -016.A 3 --008. A 4 - 003584.A 5 + -0017024 A 6 +etc. | 

 V 5 -V 4 +0-2A + -08.A 2 + -048.A 3 --0144.A 4 --008064.A 5 + -0029568.A 6 +etc.J 



The series converges so rapidly that it is seldom necessary to go beyond the fourth 

 or fifth differences, and the last result in each case is a check on the accuracy of the 

 work. But, as it requires fresh arrangements for each short series of interpolated 

 values, it is not so satisfactory for computing a lengthy table as the method above 

 explained, though a larger number of differences is required to compensate for the more 

 rapid convergence. For isolated values, however (30), is most convenient. We may 

 proceed by successively correcting the differences in a retrograde order, correcting the 

 highest employed, if necessary, to its mean value, by adding half the next above it. 

 Thus, if five orders of difference are to be used, make A* = A 5 + ^A 6 . Then — 



