543 Royal Society. 



And the same is true of 4 terms, 5 terms, or any number of terms. 

 If 3 terms have the differences reversed, the numbers added are — 



4 



If 4 terms 8 8 



If 5 terms 12 16 12 



If 6 terms 16 24 24 16 



If 7 terms 20 32 36 32 20 



&c. &c. &c. 



The law under which these numbers are formed is obvious enough. 

 The same numbers exactly are to be added to the other series (1,3, 

 9, 19, &c.) to produce the same result. 



012345 6 7 

 If the 2 series be blended together, thus 1 1 3 5 9 13 19 25, 

 &c., the differences will be 2, 2, 4, 4, 6, 6, 8, 8, &c. ; and if an odd 

 number of terms be taken (so as to begin and end with a number 

 from the same scries), and the differences be added inverso ordine, a 

 similar result occurs. Take 1 1 terms. 



1 3 5 9 13 19 25 33 41 51 61, and add the differences "in- 

 verso ordine," the series becomes, with its indices and roots, — 

 12 3 4 5 



I 11 21 29 37 



0,0,0,1 +1,0,1,3 -1,0,4,2 0,2,3,4 +1,0,0,6 6 



-1,2,4,4 43 (middle term.) 



II 10 9 8 7 -3,3,4,3 

 61 59 57 53 49 



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



-2,2,4,5 



In this case the additions are 0, 8, 16, 20, 24, 24, 20, 16, 8, ; 

 and if these be added to any other consecutive 11 terms (the 1st 

 term having an odd index), they produce the same effect as if the 

 differences were reversed ; and the resulting numbers have the pro- 

 perty of the terms equidistant from the centre, being connected by 

 their roots, having the relation so frequently mentioned. It may be 

 further remarked, that the numbers produced by reversing the dif- 

 ferences are the initial numbers from which, by adding 2, 2, 4, 4, &c., 

 6 1 may be formed of the squares, which make the differences of its 

 exterior roots 10, 9, 8, &c. 



123456789 10 



11 13 15 19 23 29 35 43 51 61 



22446688 10 

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

 12 3 456789 



21 23 25 29 33 39 45 53 61 



22446688 

 0,2,4,1 -4,2,4,5 



and so of all the others. 



The matter referred to in this Postscript tends to strengthen the 

 suggestion already made, that the pro])erties of numbers referred to 

 are connected with the mysterious and abstruse properties to which 

 Fermat referred as enabling him to prove the theorem he announced 

 of the Polygonal Numbers. 



I 



