74 Royal Society. 



and z a number prime to it (which every number not divisible by it 

 is), this is Fermat's theorem, and the author has given a new proof 

 of it. But the theorem is true though A be not a prime number, 

 provided z be prime to A and m be the number of primes to A, 

 and less than it ; and instead of 1 , any other number prime to A 

 raised to the mi\\ power may be substituted : and z'^—if"^ will be di- 

 visible by A, provided z and y be primes to A, and m be the number 

 of primes to A and less than it. 



The author has therefore in this paper offered a proof of Gauss's 

 theorem, and proved that it applies in certain cases to one half of the 

 primes, and in all cases, with certain modifications, has shown that 

 a similar property belongs to the product of the odd numbers, and 

 also of the even numbers which precede any prime number ; and 

 lastly, has shown the intimate connexion between Wilson's theorem 

 and Fermat's, and shown that each is but a part of a much more 

 general proposition, which, he observes, may itself turn out to be 

 part only of a still more universal one. 



In a postscript, the author has shown that the well-known law of 

 reciprocity of prime numbers is an immediate corollary from his 

 theorem ; and that it may be extended thus : if A and B be any 

 two numbers (not prime numbers but) prime to each other, and the 

 primes to A, and less than it, are (in) in number, and the similar 

 primes to B are (?^), then (A"—!) is divisible by B, and (B""— 1) is 

 divisible by A. 



" On the reabsorption of the Mixed Gases in a Voltameter." By 

 Professor M. H. Jacobi, in a letter to Michael Faraday, Esq., F.R.S. 

 Communicated by Dr. Faraday. 



The author found that if the mixed gases developed from the 

 decomposition of water by a voltaic current, be allowed to remain 

 in the voltameter in which they were collected, in contact with the 

 fluid which produced them, they by degrees diminish in volume, 

 and ultimately disappear by being absorbed by the fluid. He has 

 not yet fully determined the precise conditions on which this phe- 

 nomenon depends ; but he is inclined to think that it is owing to a 

 portion of the mixed gases, diffused throughout the whole liquid, 

 comipg into contact with the platinum plates, and being recombined 

 on the surface of those plates ; and this process being renewed with 

 every fresh portion of the gases which takes the place of the former, 

 the whole of the gases are thus reconverted into water. 



March 4. — " Researches into the effects of certain Physical and 

 Chemical Agents on the Nervous System." By Marshall Hall, M.D., 

 F.R.S., &c. 



The professed object of the author, in the present paper, is " to 

 detail the results of an investigation of the phenomena and the laws 

 of production and action of certain secondary or induced conditions 

 of the nervous system, which are effected by a voltaic, and proba- 

 bly by any other electric current, but persistent after the influence 

 of that current is withdrawn." This condition he designates by the 

 new term electrogenic, as describing at once the origin and the inde- 

 pendence of that condition. On the present occasion he confines 



