245 



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

 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., 

 61 may be formed of the squares, which make the differences of its 

 exterior roots 10, 9, 8, &c. 



12345 67 89 10 

 11 13 15 19 23 29 35 43 51 61 



2244668 8 10 

 0,0,3,1 0,0,3,2 -1,0,3,3 -2,0,3,4 -3,0,3,5 -4,0,36 



123456789 

 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 properties 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. 



III. " Observations on the Mer de Glace." Part I. By JOHN 

 TYNDALL, Ph.D., F.R.S. &c. Received June 10, 1858, 



(Abstract.) 



In this paper the author communicates the first part of a series 

 of observations upon the Mer de Glace, made during a residence of 

 six weeks at the Montanvert last summer *. He corroborates the 

 laws regarding the swifter flow of the central portions of the ice- 

 stream, first established by Prof. Forbes, and shows how the velo- 

 city changes as the width of the glacier varies. The Mer de Glace 

 moves through a valley which twice turns a convex curvature to 

 the east, and once to the west. The points of swiftest motion at 

 these curves are found to be not central, but thrown to that side 



* During the whole of which period he was most ably assisted by his friend 

 Mr. T. A. Hirst. 



