of a Liquid Mass without Weight. 365 



be done by help of an appropriate artifice, the transformation 

 becomes more and more rapid. For the limiting ratio 3*14, the 

 duration of the phenomenon was eleven minutes; with the ratio 

 3*18 it was only four minutes; with the ratio 3*3 it was two 

 minutes; and with the ratio 3*6 one minute. Now it is plain 

 that a more rapid transformation implies forces of greater in- 

 tensity. 



It follows, besides, from the calculation of which I have spoken, 

 that if the ratio of the distance between the bases to their dia- 

 meter is less than 7r, and if we suppose one half of the figure to 

 be slightly increased and the other half to be slightly diminished 

 in width, the capillary pressures of the swollen portion will over- 

 come those of the contracted portion, and so the cylindrical form 

 tends to reproduce itself. Lastly, the same calculation shows 

 that if the ratio is exactly equal to ir, and if the swelling and the 

 contraction are of infinitely small amount, the capillary pressures 

 are the same in both parts. We have accordingly here a second 

 method which gives the precise limit of stability of the cylin- 

 der ; only it assumes a priori that at the commencement of the 

 transformation the meridian line is a curve of sines. 



I next pass to the unduloid. Here the conditions of stability 

 are different according as the middle of the figure produced is 

 occupied by a constriction or by a bulgiug portion : in the former 

 case these conditions seem to vary according as the unduloid 

 differs more or less from a cylinder ; but in the second case the 

 limit is distinctly recognizable. A partial unduloid with a bulge 

 at the middle is exactly at its limit of stability when its bases 

 coincide with the circular sections of the necks of the two con- 

 strictions between which the bulging portion is contained. I 

 draw this conclusion both from experiment and from a course of 

 reasoning which would take up too much room for insertion 

 here. M. Lindelof has arrived at the same result mathematically. 



Hence we get another method still by which we can rigo- 

 rously determine the limit of stability of the cylinder. It is well 

 known that M. Delaunay has proved that the meridian lines of 

 surfaces of revolution of constant mean curvature are generated 

 by one of the foci of a conic section which rolls upon the straight 

 line constituting the axis of revolution. In the case of the un- 

 duloid, the rolling conic section is the ellipse ; and it is plain that 

 the portion of the described line which is comprised between 

 any two consecutive points of minimum distance from the axis, 

 corresponds to one complete rotation of the ellipse. Hence the 

 partial unduloid generated by this portion (that is to say, the 

 unduloid at its limit of stability) has a length equal to the peri- 

 )hery of the ellipse in question ; now, when this ellipse becomes 



circle, the unduloid becomes a cylinder ; and consequently this, 



