243 



one 2 more, the other 2 less, and the differences increase by 2 as the 

 terms are more distant from the middle term. 



If the number of terms be 8, the resulting series with its indices 

 and roots will be 



135 7 9 11 13 15 



1 29 53 73 89 101 109 113 



0,0,0,1 0,2,4,3 -2,2,6,3 1,0,6,6 -2,0,6,7 -5,2,6,6 -5,2,4,8 -7,0,0,8 



+2,0,0,5 -1,0,6,4 +1,2,2,8 0,2,2,9 -4,0,6,7 -3,0,0,10 



+ 1,0,4,6 -2,0,4,9 



+2,0,0,7 -1,0,0,10 



and the differences of the exterior roots will be 1, 3, 5, 7. The reason 

 of these results is, that the equidistant terms are always equal to the 

 original corresponding term in the series increased by the same 

 number. 



Thus, in the first example, if to the terms 



1, 3, 9, 19, 33, 51, 73 there be added 



0, 20, 32, 36, 32, 20, 0, the result is 



1, 23, 41, 55, 65, 71, 73, which is the 



series with the differences added " inverso ordine" And in the last 

 example, if to 



1, 5, 13, 25, 41, 61, 85, 1 1 3 there be added 



0, 24, 40, 48, 48, 40, 24, 0, the result is 



1, 29, 53, 73, 89, 101, 109, 113, that is, 



the series arising from the differences being added " inverso ordine" 

 It is worthy of observation that these numbers, 0, 24, 40, 48, 

 48, 40, 24, 0, which, added to the first 8 terms, produce a series 

 identical with the result of the differences being added "inverso 

 ordine" have the same effect upon any other consecutive 8 terms of 

 the series. Take the 2nd term as the 1st of 8 terms 



5, 13, 25, 41, 61, 85, 113, 145, to these add 

 8 12 16 20 24 28 32 



24 40 48 48 40 24 0, the result is 



5, 37, 65, 89, 109, 125, 137, 145, 

 32 28 24 20 16 12 8 



