WITHDRAWN FROM THE ACTION OF GRAVITY. 253 



of the radius r over half the chord of the small circular segments, which half 

 chord, in accordance with what we have stated, is equal to ~r. Thence we 



get A — c = ^, whence c = ,i — —r, and we have only to substitute this value 

 in the preceding formula to obtain the true theoretical value of h. The thick- 

 ness of the wire forming my rings is 0.74 millimeters ; hence — r = 0.18 milli- 

 meters, which gives as the true theoretical height of the segments under these 

 circumstances, • 



7i = 9.46 millimeters. 

 I may remark that it is difficult to distinguish in the liquid figure the precise 

 limit of the segments, i. c, the circumferences of contact of their surfaces with 

 those of the rings. To get rid of this inconvenience, I measured the height of 

 the segments, commencing only at the external planes of the rings; i. c, in the 

 case of each segment, commencing at a plane perpendicular to the axis of revo- 

 lution, and resting upon the surface of the ring on that side which is opposite 

 the summit of the segment. The quantity thus measured is evidently equal to 

 the total height minus the versed sine of the small circular segments which we 

 have considered above ; consequently these small circular segments being simi- 

 lar to that of the spherical segment, we obtain for the determination of this 



versed sine, which we shall denote by^i the proportion — = — , which in the 



—r 

 2 



case of our liquid figure gives f=0.Q5 millimeters, whence 



h — y=: 9.41 millimeters. 

 This, then, is definitively the theoretical value of the quantity which was required 

 to be measured. 



41. Before pointing out the process which I employed for this purpose, and 

 communicating the result of the operation, I must preface a few important 

 remarks. If the densities of the alcoholic mixture and of the oil are not rigor- 

 ously equal, the mass has a slight tendency to rise or descend, and the height 

 of one of the segments is then a little too great, whilst that of the other is a 

 little too small ; but we can understand that if their difference is very small, an 

 exact result may still be obtained by taking the mean of these two heights. 

 We thus avoid part of those joreliminary experiments which the establishment 

 of perfect equality between the two densities requires. But one' circumstance 

 which requires the greatest attention is the perfect homogeneity of each of the 

 two liquids. If this condition be not fulfilled with regard to the alcoholic mix- 

 ture, i. e., if the upper part of this mixture be left containing a slightly greater 

 proportion of alcohol than the lower portion, the liquid figure may appear 

 regular and present equal segments ; all that is required for this is, that the 

 mean density of that part of the mixture, Avhich is at the same level as the 

 mass, must be equal to the density of the oil ; but imder these circumstances 

 the level of the two segments is too low. In foct, the oil forming the upper 

 segment is then in contact with a less dense liquid than itself, and, conse- 

 quently, has a tendency to descend, whilst the opposite applies to the oil form- 

 ing the inferior segment.* Heterogeneity of the liquid produces an opposite 

 efiect, i. e., it renders the height of the segments too great. In fret, the least 

 .dense portions rising to the upper part of the mass tend to lift it up, whilst the 

 most dense portions descend to the lower part, and tend to depress it. Now, 



* By intentionally producing very great heterogeneity in the alcoholic mixture, (§ 9 of the 

 preceding memoir,) and employing suitable precautions, a perfectly regular cylinder may be 

 formed, the bases of which are absolutely plane. 



