636 PROCEEDINGS OF THE AMERICAN ACADEMY 



These two equations must be solved as simultaneous. We have been 

 able to do this by the aid of a set of definite integrals given in the appen- 

 dix of Kramp's Analyse des Refractions Astronomiques et Terrestres, 

 published at Strassburg in 1799. Kramp's table contains values of the 

 integral 



for 301 values of a from to 3.00. This table shows, in the first place, 

 that various values of 



Xoo 



e dl3. 



obtained on the assumption that Ir is of the order of magnitude of the 

 diffusion constant of an electrolyte, and that — ^ is of the order of magni- 

 tude given by our experimental data, differ from each other and from 

 V^r by less than 2 per cent. This means merely that the change of 

 concentration, v, of the substance originally in the gelatine can be 

 neglected. 



Dividing equation (11) by equation (10), and extracting the square 

 root, we have 



/-»00 



/ ^ 



t/.0153 



,0153 

 2ii 



J.o: 



'^e-^'d(3 



0167 

 2a 



= 2. (12) 



To solve this equation for a it is only necessary to find two values in 

 Kramp's table for which the lower limits of integration are in the ratio 

 0.0167 to 0.0153, and for which the value of the integral corresponding to 

 the second is double the value of the integral corresponding to the first. 



To give an idea of the procedure in such a calculation we shall give on 

 the next page a few values from Kramp's table in the neighborhood of 

 the solution, of which the starred value is seen to be the one satisfying 

 equation (12). 



The ratio of 0.0167 to 0.0153 is 1.091. 



a and ai are the lower limits of integration, A and A^ the correspond- 

 ing values of the integral. The correct solution is singled out by the 



relation —t- = 2, 

 Ai 



