112 



MR J. Y. BUCHANAN ON THE 



When the solution does not follow this law quite exactly, let the displacement for 

 any particular value of m be A^ = Ai ; then the degree in which the solution conforms 

 to the law is indicated by the difference x —m. 



In the table, the displacements given in column 6 are calculated on the basis of 

 the second hypothesis. For them, therefore, the relation A^ = A^™ holds good, and the 

 value of m (column 1) for any solution expresses not only its molecular concentration, 

 but also the exponent of its displacement, that of A^ being taken as unity. 



The values of the displacement of the solutions in column 4 of the table are arrived 

 at on the basis of our first hypothesis ; consequently any value of A in this column is 

 given by the equation A,,^ = 1 -000 + 0"04m. But none of the solutions dealt with in this 

 memoir follow this law at all concentrations, though some of them approximate to it at 

 high concentrations. It is therefore of use, in order to augment the illustrative value 

 of the table, to determine the exponents of the values of A in column 4 when referred 

 to that of A = 1 -020 for m = 1/2 as 1/2. This has been done, and the results are entered 

 in the following table : — 



m. 



X. 



x-m. 



m. 



1/2 



X. 



l/m-1/x. 



10 



8-495 



- 1-505 



1/2 



0-00 



9 



7-76 



-1-24 



1/4 



1/3-98 



+ 0-02 



8 



7-00 



-1-00 



1/8 



1/7-943 



+ 0-057 



7 



6-23 



-0-77 



1/16 



1/15-873 



+ 0-127 



6 



5-43 



-0-57 



1/32 



1/31-706 



+ 0-294 ■ 



5 



4-60 



-0-40 



1/64 



1/63-412 



+ 0-588 



4 



3-75 



-0-25 



1/128 



1/127-52 



+ 0-480 



3 



2-86 



-0-14 



1/256 



1/254-06 



+ 1-940 



2 



1-94 



-0 06 



1/512 



1/508-90 



+ 3-100 



1 



0-99 



-0-01 









1/2 



0-50 



000 









§ 45. In the following tables the displacements of most of the solutions have been 

 treated along these lines. In the first four tables we have for the solutions of the salts 

 of the ennead MR the values of x and of x—m for the strong solutions, and those 

 of X and 1/m - l/x for the dilute solutions. For solutions of other salts the tables give 

 only the values of x —m or 1/m - l/x, as they are sufficient. 



The numbers representing the values of x and m for the strong solutions are the 

 numerators of vulgar fractions having unity for common denominator. The numbers 

 representing the values of l/x and 1/m for the weak solutions are the denominators of 

 vulgar fractions having unity for common numerator. The measure of the departure of 

 the displacements of solutions of a particular salt from the geometric law of the second 

 hypothesis is found for the strong solutions in the column headed {x — m). For the weak 

 solutions the corresponding column is headed (l/m — l/x), so that the signs prefixed to 

 the numbers in these columns mean the same thing in both tables : — the + sign means 

 that x>w, the — sign that x<jn. In the strong solutions x = m when m= 1, and the 

 remaining values of m increase; in the weak solutions x = m when m=l/2, and the 

 remaining values of m diminish. 



