Baron Pollock on some Properties of Numbers. 539 



and the lower terms have as indices (4 — 1), (4 — 2), &c., the series 

 with its indices will be 



2 3 4 5 6 



9 13 17 31 25 



and if the terms less than the middle term be di\'ided into 4 squares 

 with exterior roots having the differences indicated by their respect- 

 ive indices thus, 



2 3 4 



9 13 17 



0,1,2,2 -1,2,2,2 -2,0,3,2 



then the terms greater than« the middle term will have this relation 

 to the terms less than the middle term ; the terras equidistant from 

 the middle term will have their middle roots the same, and the dif- 

 ferences of the exterior roots will increase ; those nearest the middle 

 term will have a difference of 1, the next 2, and so on, thus: 



2 3 4 5 6 



9 13 17 21 25 



0,1,2,2 -1,2,2,2 -2,0,3,2 -2,2,2,3 -2,1,2,4 



An algebraic proof is then given as to a series whose middle 

 term is 71 and common difference p ; and as n may be odd or even, 

 and^ also, and the index of differences may be ininus as well as ^j/m«, 

 the theorem applies frequently to even numbers, but not universally. 

 The following example is given of the theorem applied to 1 7 terms 

 of a series whose first term is 25, and common difference 1 : — 



_7 _6 -5 -4 -3 -2 -1 

 25 26 27 28 29 30 31 32 



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



33 

 98765432 0,4,4,1 

 41 40 39 38 37 36 35 34 



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



Comparing the terms above with the terms below, it is manifest 

 the terms of the series are divisible into 4 squares whose roots con- 

 form to the law of the theorem. It is then shown that the odd squares, 

 and also all the numbers mentioned in the beginning of the j)aper, 

 can be made terms in an arithmetic series, and will therefore have 

 the property stated. It is then suggested that the properties of 

 numbers stated in the paper may have been in some form a portion 

 of the mysterious ])roperties of numbers by which Fcnnat {innounced 

 he could prove his celebrated theorem of the polygonal numbers. 



A Postscript was added, dated 20th May, which is here given 

 entire. 



Since this pa[)er was sent to the Society, some other theorems of 

 a similar kind have occurred to me, in which the terms of a series 



