Baron Pollock on some Properties of Numbers. 541 



Here there is a middle term ; the equidistant terms have the same 

 middle roots, the exterior roots are (next to the middle term) the 

 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 nuniber of terms be 8, the resulting series with its indices 

 and roots will be — 



13 5 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 -7 008 

 +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 teritji' 



1, 3, 9, 19, 33, .51, 7'3. Uierc be added 



0, 20, 32, 36, 32, 20, QJ-.fl^e result is 



1, 23, 41, 55, 05, 71, 5:3; which is the 



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

 example, if to .,; . 



1, 5, 13, 25, 41, 61, 85,^ 113 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 IG 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 



in which last series the differences are reversed or added " inverso 

 ordine" The appropriate roots of these numbers are — 



3 5 7 9 11 13 15 17 



5 37 65 89 109 125 137 145 



-1,0,0,2 -1,2,4,4 -3,2,6,4-2,0,6,7-3,0,0,8 -6,2,6,7-6,2,4,9-8 9 



+ 1,0,0,6 -2,0,6,5 0,2,2,9 -1,2,2,10 -5,0,6,8 -4,0,0,11 



0,0,4,7 -3,0,4,10 



+ 1,0,0,8 -2,0,0,11 



which may be immediately obtained from the former series, the 

 middle roots Iieing the same ; and the exterior roots, one of them 

 one less, the other one more. In tiiis way any consecutive 8 terms, 

 with the differences reversed, may be each divided into 4 squares 

 throughout the whole series. 



