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LXIV. On an extension of Sir John WilsorCs Theorem to all 

 numbers whatever. Bi/J. J. Sylvester, Professor of Natu- 

 ral Philosophy in University College, London. 



To the Editors of the Philosophical Magazine and Journal, 

 Gentlemen, 



T^HE annexed original theorem in numbers will serve as a 

 -*- pendant to the elegant discovery announced by the ever- 

 to-be-lamented and commemorated Horner*, with his dying 

 voice, in your valued pages. 



Theorem. 

 If N be any number whatever and 



Pp P'i^ Vs Pc 



be all the numbers less than N and prime to it, then either 



i'l • i^2 • Ps Pc+ 1 



orelse Pi - P^ - P3 Pc — ^ 



is a multiple of N. 



Yours with high respect, 

 University College, London, Oct. 22, 1838. J. J- SYLVESTER. 



LXV. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



June 21. — The following papers were read, viz. 



" On the structure of the teeth, the vascularity of those organs, 

 and their relation to bone." By John Tomes, Esq. Communicated 

 by Thomas Bell, Esq., F.R.S., Professor of Zoology in King's Col- 

 lege, Londdin. 



The microscopical examinations which the author has made of the 

 structure of the teeth of man and various animals, lead him to the 

 conclusion that their bony portions are formed of minute tubes, dis- 

 posed in a radiated arrangement, in lines proceeding everywhere 

 perpendicularly from the inner surface of the cavity containing the 

 pulp. These tubuli are surrounded by a transparent material, which 

 cements them together into a solid and dense mass. He finds, by 

 applying the test of muriatic acid, that carbonate as well as phos- 

 phate of lime enters into their composition. In man, the tubuli, du- 

 ring their divergence from their origin at the surface of the central 

 cavity, send off a number of very minute fibrils ; and on. approach- 

 ing the enamel or the granular substance, which cover respectively 

 the crown and the fangs of the tooth, the tubuli divide into smaller 



* Horner*s proof is highly valuable as a novel and highly ingenious form 

 of reasoning, but his theorem may be deduced with infinitely more ease and 

 brevity from Fermat's than he seems to have been aware of. 



