On Staudt's Theorems of Polygons and Polyhedrons. 335 



Silica 0-818 



Phosphoric acid 0*500 



Nickel (and cobalt) .... 0-129 



Lime and magnesia . . . . 0-178 



Alumina 0-213 



Vanadic acid trace 



Manganese trace 



Protoxide of iron (by difference) 98*171 



100-009 

 The residue, insoluble in very dilute nitric acid, contained 

 also portions which obeyed the action of the magnet ; and these 

 in like manner were subjected to analysis, and gave the following 

 results calculated in the state of oxides : — 



Silica 0-619 



Phosphoric acid 3-159 



Vanadic acid 1-402 



Oxide of nickel (with oxide of cobalt) 0-737 



Oxide of iron 94-464 



Manganese trace 



100-381 

 So far M. Bahr, the original source of whose paper I am 

 unable to state, as it is not given in the Spanish version of it. 

 In conclusion 1 may observe, that I possess in my collection of 

 minerals a specimen of limonite from Passau in Bavaria contain- 

 ing small threads and veins of iron in the metallic state. 

 I am. Gentlemen, 



Your obedient Servant, 



W, G. Lettsom. 



LIII. On Staudt's Theorems concerning the contents of Polygons 

 and Polyhedrons, with a Note on a new and resembling Class of 

 Theorems. By J. J. Sylvester, Barrister-at-Law *. 



THE beautiful and important geometrical theorems of Staudt 

 arc, I believe, little, if at all, known to English mathema- 

 ticians. They originally appeared in Crelle's Journal for the 

 year 1843, and have been recently reproduced in M. Terquem^s 

 Nouvelles Annates for the August Number of the present year. 



These theorems may be summed up, in a word, as intended to 

 show the possibility and method of expressing the product of any 

 two polygons or any two polyhedrons as entire functions of the 

 squares of the distances of the angular points of the two figures 

 from one another. The well-known expression for the square of 

 * Communicated by the Author. 



