424 [Jan. 29, 



the completed series of integrals of the first equation. The reader 

 is reminded that by the completed series of integrals is meant, not all 

 the integrals of the first partial differential equations that exist, but 

 all that arise from a certain root integral by a certain process of deriva- 

 tion, together with the root integral itself. Now the answer here to 

 be established to this inquiry is the following. The first of the par- 

 tial differential equations necessarily will, and others may, be satisfied 

 by the proposed function irrespectively of its form. If the number 

 of equations of the completed system which is not thus satisfied is 

 odd (this is the condition in question), the form of the function which 

 will satisfy all is determinable by the solution of a single differential 

 equation of the first order, capable of being made integrable by means 

 of a factor. 



Although the direct subject of this paper is the solution of partial 

 differential equations of the first order, I wish it rather to be received as 

 a slight contribution to that theory of the dynamical equations which 

 was first published in the Philosophical Transactions, and which sug* 

 gested to Jacobi the line of investigation which I here only seek to 

 pursue a little further. 



January 29, 1863. 



Dr. WILLIAM ALLEN MILLER, Treasurer and Vice- 

 President, in the Chair. 



Mr. Henry John Carter was admitted into the Society. 

 The following communication was read : 



" On the Absorption of Gases by Charcoal. No. I." By Dr. R. 

 ANGUS SMITH, F.R.S. Received December 27, 1862. 



(Abstract.) 



The following is a summary of the author's observations : 



1 . Charcoal absorbs oxygen eo as to separate it from common air, 

 or from its mixtures with hydrogen and nitrogen, at common tempe- 

 ratures. 



2. Charcoal continues the absorption of oxygen for at least a month, 

 although the chief amount is absorbed in a few hours, sometimes in 

 a few seconds, according to the quality of the charcoal. 



