408 PRACTICAL MATHEMATICS 



Hence from the term u there can now be taken two different 

 sets of successive differences. There is the first set Aw , A 2 w , 

 A 3 w , etc., running diagonally downwards, and from these values 

 we are enabled to find the value of any term below w , for it has 

 been already shown that u n = (1 + A) n w . 



There is the second set of differences 8w , S 2 M , S 3 w , etc., 

 running diagonally upwards, and from these values it should be 

 possible to find the value of any term above w . 



Now w_j = u Q Sw ............... (1) 



M_ 2 = W-! Sw_j 



= u Sw (Sw S 2 M ) 



= w - 2Sw + S 2 w ............ (2) 



W_ 3 = U- 2 Sw_ 2 



= UQ - 2SW + S 2 M - (8W - 28 2 W + S 3 W ) 



= U Q - 3Sw + 38 2 w - 8 3 w ......... (3) 



W_ 4 = M_ 3 - 8W_ 3 



= U Q - 38w + 3S 2 w - S 3 w - (Sw - 3S 2 w + 3S 3 w 



= W - 4Sw + 68 2 w - 4S 3 M + 8X ...... (4) 



10S 2 w - 10S 3 w + 5S 4 w - S 5 % ... (5) 



The multipliers of the differences are evidently the same as the 

 binomial coefficients in the expansions for which the powers are 

 1, 2, 3, 4, and 5 respectively, but the signs are alternately positive 

 and negative. 



n(n 1) .. n(n \}(n 2) ... 

 Hence w_ n = u - n 8u + , *$ 2 u - -* - ^ - '- S 3 w + . . . 



IL l. 



Symbolically, then, 



t_ 4 = (1 -4S + 6S 2 -4S 3 + 8 4 )w = (1 



w_ 5 = (1 - 5S + 10S 2 - 108 3 + 5S 4 - 8 5 )w = (1 -S) 5 w 



,, n(n !). n(n l)(n 2)^. 

 72 



.)w = (l-S) Mfl 



In which (1 S) 2 w means that (1 8) operates on w twice, or 

 more generally (1 8) n w means that (1 8) operates n times 



On U. 



