136 Mr. S. U. Pickering on the Densities 



that the arrangement of points exhibited by the diagrams is 

 purely accidental. 



Setting aside for a moment the concordance of independent 

 results, and thus ignoring the main grounds of mv conclusions, 



/ COO - / 



let us see what Prof. Pucker's results do towards disproving 

 my opinions in the one particular case which he investigates. 



I may as well state at once that I consider Prof. Pucker's 

 equation to agree with the experimental results just as satis- 

 factorily as my own drawings do, and more satisfactorily 

 perhaps than he asserts ; for I have recently revised my esti- 

 mate of the experimental error, and obtain a value for it 

 somewhat larger than I previously did. From all my deter- 

 minations done in duplicate with water I get "000012 ('0003 

 gram on the 25 cubic centiin. taken) as the mean error of a 

 single observation, and. by a graphic method, described by me 

 in the Ber. d. deutsch. chem. Gesel. (xxiv. p. 3332), applied to 

 the results with sulphuric acid itself, 1 get -000011. This 

 agrees perfectly with my original drawing, which attributed 

 an apparent error of "000013 to the experimental points (due 

 allowance being made, of course, for the fact that these were 

 " differential" points, of which the error would be sometimes 

 greater and sometimes less than that of a single determination, 

 according to the magnitude of the actual differences taken), 

 and equally well with Prof. Pucker's equation, which gives 

 •000012. 



Now, as is well known, any figure, however complicated, 

 may be expressed by an equation, within any assigned limits 

 of accuracy, even of the simple form y = a + bx + cx 2 . . . + zx n , 

 provided a sufficient number of terms be introduced into it, 

 and this is true even if the available points form in reality 

 two distinct parabolas or other curves. This Prof. Pucker 

 himself points out to be the case (p. 305) ; so that the mere 

 fitting on of an equation is no proof of continuity. The task 

 of finding an equation to fit fairly well on to a figure made 

 up of such independent curves is naturally more simple when 

 these curves meet almost " end on," if I may use such an 

 expression, and show no changes of so marked a character as 

 to be correctly described as violent and sudden, or as present- 

 ing any " awkward corners." Such is the nature of that 

 portion of my density-differential curve selected by Prof. 

 Pucker for examination ; the changes of curvature which I 

 suppose it exhibits are, as he admits, " minor changes," and 

 by no means so clearly indicated as some of those in other 

 parts of the figure ; and in such a figure it would evidently 

 be possible, by taking mean points between each pair of expe- 

 rimental points, to get an equation, even of the simple para- 



