of Solutions of Soda and Potash, 361 



Examination of the Results with Soda. 



The results with soda are given in Table I. A considerable 

 number of the observations were repeated with solutions of 

 nearly the same strengths, and the means deduced from these 

 duplicates have been used for plotting purposes. From this 

 cause the mean experimental error is less than it was in the case 

 of my determinations with sulphuric acid ('000011, see Phil. 

 Mag. xxxiii. p. 136). The graphic method — which is based 

 on measuring the errors of every two alternate points as com- 

 pared with arcs of circles drawn through three other alternate 

 points (Ber. d. deutsch. chem. Gesell. xxiv. p. 3332, xxv. 

 p. 1100) — gave '00000734 when applied to the whole series 

 of values ; but if there are any breaks in the figure it is 

 obvious that the arcs should not be applied so as to bridge 

 over a break, as the errors in its neighbourhood would thereby 

 be made to appear larger than they really are. When applied 

 so as to avoid bridging over the breaks which the figure 

 appears to show, the mean error was found to be "00000631 ; 

 and, as the following examination sufficiently establishes the 

 reality of these breaks, this value for the error has been 

 accepted. The general character of the results, however, it 

 must be noted, would not be altered if the higher estimate 

 were taken. 



The densities themselves, when plotted against percentage 

 composition, form a figure which shows that throughout 

 they increase with the strength at a decreasing rate. In 

 order that a scale might be adopted sufficiently open to 

 give the experimental error a visible magnitude, the differ- 

 ences between the densities and a straight line (third column), 

 or between the densities and a simple parabola (fourth column), 

 have been used for the examination. The values given by the 

 former are preferable for determining the position of any 

 breaks, owing to the simpler character of the figure formed 

 by them ; those given by the latter are preferable for the 

 estimation of the errors of the drawings, owing to the more 

 open scale which can be used in plotting them. These latter 

 values are shown in the diagram, though the scale which has 

 necessarily been adopted there is too restricted for purposes 

 other than those of illustration. 



A glance at this figure will be sufficient to show that they 

 cannot be expressed by any single bent-lath curve ; and a 

 more careful examination of them with the lath indicates the 

 probable existence of as many as eight changes of curvature, as 

 shown in the diagram (p. 363). When the apparent errors of 

 the points according to such a drawing are measured, the results 



