On the Analysis of Waters by the Soap-Test. 171 



r m+\ symbols can be arranged in (r m+1 — 1) : (r— 1) columns 

 of r-plets, each column containing all the symbols, and so 

 that every duad shall be once, and once only, employed. 



Theor. III. If r be any prime number, (r 2 -f r-f-1) symbols 

 can be formed into (r 2 + r + 1) r+ 1-plets, so that every duad 

 shall be once, and once only, employed. 



Theor. IV. If r be any prime number, and r 2 + ^+l be 

 prime, (n=) (r 2 + r + l)(r+l) symbols can be formed into 

 71.71 — 1 : r.r+1 r+ 1-plets, so that every duad shall be once, 

 and only once, employed. 



Theor. V. Nine symbols can be arranged in seven sets each 

 of four columns of triads, every column containing the nine 

 symbols once, and so that no triad shall be twice employed. 



Theor. VI. If R = A a B 6 C e ... L'M m N be any integer, of 

 which A, B, C, ... L, M are prime factors in decreasing order, 

 N being any number less than M, R symbols can be arranged 

 in (A a -1):(A-1) columns of A-plets, + A a .(B 6 -l):(B-l) 

 columns of B-plets, + A a B*.(C c -l) : (C-l) columns of 

 C-plets, + ... + A a B*C° ... L Z (M W -1) : (M-l) columns of 

 M-plets, +A a B b C c ...VM m columns of N-plets, and this 

 so that each column shall exhibit all the R symbols, and that 

 every duad shall be once employed and repeated nowhere. 



Croft Rectory, near Warrington, 

 August 6, 1850. 



XX. On the Action of the Soap-Test 7ipon Water containing 

 a Salt of Magnesia only, and likewise upon Water containing 

 a Salt of Magnesia and a Salt of Lime. By Dugald 

 Campbell, F.C.S. London*. 



FOR some time back I have been engaged in analysing 

 specimens of water — determining by actual experiment, 

 besides the salts of the alkalies which they contain, the amount 

 of phosphate of iron and salts of lime and magnesia in a gal- 

 lon, testing them at the same time with reference to their de- 

 grees of hardness, according to the plan adopted by Prof. 

 Clark in the specification of a patent printed in the Reper- 

 tory of Patent Inventions for 1841. 



When the quantities of phosphate of iron and salts of lime 

 and magnesia found in a gallon of water were calculated each 

 into its equivalent of chalk and added together, I had antici- 

 pated that their amount would correspond, or nearly so, with 



* Communicated by the Author, having been read before the British 

 Association, August 1850. 



