408 Mr. S. U. Pickering on the Expansion 



show irregularities too great to permit of any satisfactory 

 conclusions being drawn from them. 



Matthiessen's determinations (Pogg. Ann. cxxviii. p. 527, 

 and fig. 8) indicate a change at about 20°, but none at 

 50°-60° : they would not be sufficiently numerous to show 

 any which existed at temperatures below 20°. I have repre- 

 sented a portion of the experimental curve itself in fig. 12 : 

 the close similarity with my own as regards the change near 

 20° will be noticed, though the curves, referring as they do to 

 weights and volumes respectively, differ as to their position. 

 Rosetti's experimental curve is of a precisely similar nature. 



Jolly's values (Sitzungsber. d. Akad. zu Miinchen, 1864, (i) 

 p. 141) give a differential represented by fig. 9. They indi- 

 cate a change at about 50°, but they would not be sufficiently 

 numerous to show any changes at lower temperatures. Jolly 

 quotes some results obtained by Henrici, but they are not 

 sufficiently numerous for the present purposes. 



Hagen's determinations (Abhandl. d. k. Akad. d. Wissen- 

 schaft. z. Berlin, 1855; Math. Abhandl. i.) are very numerous, 

 and for the purposes of direct differentiation must be arranged 

 in two series. They indicate cbanges at 50°, 20°, and 10°, 

 but the exact form of the figure between 20° and 10° is 

 doubtful : the changes at these two temperatures depend on 

 the fact of the curves above and below them ceasing to be 

 applicable below and above the two temperatures respectively. 

 The points given by the second series into which his results 

 were divided have not been inserted here owing to lack of 

 room : they do not extend above 34 . Hagen's values refer 

 to actual weights ; they have been multiplied by f so as to 

 reduce them to about the same relative magnitude as those 

 results which are expressed as volumes or densities. 



None of these results, if taken separately, would be sufficient 

 to prove the changes which they indicate, but when taken 

 together they certainly afford sufficient evidence for the asser- 

 tion that these changes are highly probable. The evidence 

 may be summarized as follows : — 



Change at Indicated by the results of 



50°-60° . . Hagen, Jolly, Kopp, Pierre, and Rosetti. 



17°-20° . . Hagen, Kopp, Matthiessen, Pickering, 

 Pierre, and Rosetti. 

 9°-ll° . . Hagen, Kopp, Pickering, and Pierre. 

 r,o p- ( Despretz's results showing some 



oo.c p- \ irregularity at these low tem- 



(_ peratures. 



Such concordance in the results of different observers can 



