428 Mr. S. U. Pickering on the 



differentiating them; the differentiation being performed 

 either on the experimental values themselves or on the smooth 

 curves representing them, generally on both. 



For a direct differentiation the difference between the values 

 obtained for the densities, or any other properties, of two solu- 

 tions was divided by the difference in their percentage com- 

 position. The result gave the first differential coefficient; that 

 is, the rate of change in density between these two percent- 

 ages, or, as it has to be less perfectly expressed, at a point 

 intermediate between the two. If the densities themselves 

 form a continuous and regular curve, the differential deduced 

 from them will form either another continuous curve or a 

 straight line ; whereas, if the densities are represented by 

 several different curves, the differential will consist of as many 

 other curves or straight lines. Thus, differentiation was used 

 primarily as a means of rendering more apparent any sudden 

 changes of curvature in the figure representing the experi- 

 ments ; but it generally indicated the nature of the component 

 curves of the latter, since parabolic curves of the second order 

 give straight lines as their first differential. In all cases 

 either the whole or else the greater part of the first differential 

 figures obtained were curvilinear, but, on proceeding to a 

 second differentiation, a rectilinear figure was obtained. 



The only case in which the second differentiation could be 

 performed on the first differential values themselves was that 

 of the electric conductivities ; in the other cases the magnitude 

 of the errors of the experiments approached too nearly to that 

 of the quantities constituting the second differential to render 

 such a method practicable. In such cases it was necessary to 

 reduce the error by plotting out the first differential points, 

 drawing a smoothed curve through them, and performing the 

 second differentiation on readings taken from this. Prof. 

 Arrhenius objects to such a process : but it is one which every 

 physicist uses to reduce experimental error, and objection 

 might as well be made against taking the mean arithmeticallv 

 of several determinations at the same point, as against taking the 

 mean of a curve dia grammatically from several determinations 

 along its course. To say that in such a case the mean value 

 taken, or the mean curve drawn, " entirely lacks experimental 

 foundation/'' and still more, to say that I admit that it does so, 

 is certainly incorrect. 



In all cases, except that of the densities and electric con- 

 ductivities, the first differentiation was applied to the smoothed 

 curve drawn to represent the experiments, as well as to the 

 experimental values themselves : the two first differential 

 figures thus obtained were treated separately and the results 

 compared. 



