1885.] A General Theory of Mathematical Form. 393 



(2.) The results obtained by agitating water with various solid 

 materials show that a very great reduction in the number of sus- 

 pended organisms may be accomplished by this mode of treatment, 

 and the complete removal of all organisms by agitation with coke is 

 especially worthy of notice. 



(3.) Again, the results obtained with Clark's process show that 

 we possess in this simple and useful mode of treating water a means 

 of greatly reducing the number of suspended organisms. 



(4.) Thus, although the production in large quantity of sterilised 

 potable water is a matter of great difficulty, involving the continual 

 renewal of filtering materials, there are numerous methods of treat- 

 ment which secure a large reduction in the number of organisms 

 present. 



Moreover, in judging of the value of filtering materials from exami- 

 nations of this kind, it is only reasonable that a preference should 

 be given to those materials with which a practically pure cultivation 

 is obtained in the filtrate over those materials which appear to exercise 

 no selective action upon micro-organisms. 



In conclusion, I would point out that it is very desirable that 

 experiments of this kind should be greatly multiplied and repeated 

 under varying conditions, and it is my intention to continue and 

 extend this examination. 



III. "A Memoir introductory to a General Theory of Mathe- 

 matical Form." By A. B. Kempe, M.A., F.R.S. Received 

 May 18, 1885. 



(Abstract.) 



The memoir is divided into 426 short sections which are arranged 

 under 42 heads. Each head is given in the abstract, with a brief 

 reference to the nature of the sections it comprises, except in the 

 case of the second head, viz., " Fundamental Principles," the sections 

 under which are given almost in full. 



§§ 1 — 2. Scope of the Memoir. 



The object of the memoir is the treatment of the " necessary 

 matter " of exact or mathematical thought as a connected whole ; the 

 separation of its essential elements from the accidental clothing — 

 algebraical, geometrical, logical, &c. — in which they are usually pre- 

 sented for consideration ; and the indication of that to which the 

 infinite variety which those elements exhibit is due. 



The memoir is introductory only, comprising the statement of 

 fundamental principles, and the vindication of their truth by a suffi- 

 cient variety of applications. 



