of a Liquid Mass without Weight. 363 



actually at this limit, the unduloid would coincide with it, or, if 

 the expression is preferred, it would differ infinitely little from it, 

 since the faintest trace of an enlargement and a contraction would 

 then be sufficient to bring about the spontaneous transformation. 

 Hence, when a cylinder of liquid is precisely at its limit of sta- 

 bility, we can always conceive of a partial unduloid which differs 

 infinitely little from this cylinder, and is made up of exactly one 

 enlargement and one contraction. 



Now we have the differential equation of the first order of the 

 meridian lines of equilibrium-figures of revolution, and in the 

 case of an unduloid infinitely near to a cylinder, this equation 

 can be integrated by the ordinary methods. It then gives a 

 curve of sines {sinudide) for the meridian line; and if we try 

 what is the sum of the lengths of the chords of a convex and of 

 a concave arc of the curve of sines, we find that, representing this 

 sum by L, and the radius of the cylinder by r, we have the rela- 



tion L = 27rr. 



But the length L is evidently that of the cylinder at its limit of 

 stability; and if it is divided by the diameter 2r, it gives what I 

 have called the limit of stability of the cylinder, which it will be 

 seen is exactly equal to the ratio tt. 



I have verified this result of calculation by new experiments 

 more accurate than my first, and likewise performed upon cylin- 

 ders of oil contained between disks and surrounded by the alco- 

 holic mixture. Before stating their results it is necessary to 

 make one remark. When we are not too near the limiting con- 

 dition, there are two characters which clearly indicate stability or 

 instability : if, when a cylinder has been formed approaching the 

 limit, we produce in it artificially a slight enlargement and con- 

 traction by impelling the oil with the point of the syringe, and 

 the figure then resumes its previous form, it is evident that it 

 still possesses actual stability ; on the other hand, if while we 

 are trying to produce the cylinder (that is, while there is still an 

 excess of oil and we are withdrawing some so as to arrive at the 

 cylindrical form) the figure begins to change of its own accord 



^ before this form is reached, we must conclude that the cylinder 

 which it is wished to produce would be unstable. 

 In the experiments in question, I gave in succession the follow- 

 ing values to the ratio between the distance of the disks to their 

 diameter : — 



3-6, 3-3, 3-18, 314, 3*09, 3-11, 313. 

 It will be seen that the first three ratios exceeded the limit 3*1 4, 

 but that they became successively nearer and nearer to it j now, 

 for these three ratios the above-mentioned character of instability 

 was distinctly shown, but in a decreasing degree from the first 



2B2 



