140 Mr. S. U. Pickering on the Densities 



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Prof. Pucker admits that "the curves in some parts — if not 

 discontinuous — have peculiar features which suggest special 

 physical causes : " his " doubts have always had reference to 

 the minor changes of curvature," such as those here dis- 

 cussed, " and to the use of the ruler in detecting them/' He 

 certainly does not appear, however, to have done much to 

 justify these doubts ; for however we may differ in our 

 explanations of these changes, or of the relative degree of 

 suddenness with which they occur, he must admit that I with 

 my bent ruler have as a matter of fact discovered the exact 

 points at which he with his mathematics has found that a fresh 

 order of conditions becomes appreciable, and again disappears. 



Prof. Riicker would, no doubt, point out that, although the 

 fourth term in his equation is appreciable throughout a certain 

 range only, it is not actually non-existent in other parts, and 

 that, therefore, there is no true mathematical discontinuity. 

 I never, however, ventured to assert that the changes occurred 

 so suddenly as to prove strict mathematical discontinuity ; and 

 I fail entirely to see how such discontinuity could ever be 

 proved or disproved by any experiments which were not 

 absolutely free from experimental error, and which were not 

 infinite in number. Indeed, those hydrates, the presence of 

 which in appreciable quantities conditions a particular rate 

 of change of density &c. between certain points, cannot be 

 regarded as being entirely absent from other solutions — a view 

 which the principles of dissociation and the gradual removal 

 of such hydrates or substances in the solid or gaseous form 

 from such solutions necessitate (see Chem. Soc. Trans. 1889, 

 pp. 22, 23; 1890, pp. 138, 340) — the only statement which we 

 can make on the strength of experiments is that the amount 

 of the substances present is inappreciable, or otherwise, by 

 those experiments. In the same way the suddenness with 

 which a change of curvature occurs can only be determined 

 within limits comparable with those of the experimental error. 

 My " breaks," in fact, are similar to those which we get in 

 most cases of a change of condition, where the practical 

 existence of the break is beyond doubt, although its absolute 

 abruptness may always be doubted, and could certainly never 

 be proved in a strictly mathematical sense. 



Again, it might be urged that if in a series of experimental 

 results such as the present the practical disappearance of any 

 term in a continuous equation devised to represent them 

 occurs at any particular point (as Prof. Pucker's fourth term 

 does at 72 per cent.), it would not cease to be appreciable 

 till some other point were reached if the accuracy of the de- 

 terminations were increased, say, tenfold. This is as it may be : 

 but it is useless to speculate as to what might be the case with 



