366 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
cumference of a circle having its centre on this same axis ; finally, at the limit, 
the nodes entirely disappear, and the above ares become so many demi-cireum- 
ferences, tangents one to the other. The spheres, therefore, generated by these 
semi-circumferences are tangents also, and hence it results that one of the limits 
of the variations of the nodoid is an indefinite series of equal spheres, which 
touch each other upon the axis. We already know (§ 8) that a similar series 
of spheres constitutes one of the limits of the variations of the unduloid, so 
that this limit is common to the two figures, and forms consequently the transi- 
tion from one to thc other; this is likewise shown by the experiments of § 31, 
since, in passing from the cylinder to the portion of a sphere, the figure realized 
always pertains to the unduloid. The meridian line of a nodoid, but little re- 
mote from the limit just ascertained, is represented by Fig. 39. 
#ug. 39 
§ 35. The variations of the nodoid have a second and very remarkable limit. 
Let us realize, by the process explained in § 27, the portion of a nodoid gene- 
rated by an isolated node; let us suppose, moreover, that we successively repeat 
the experiment with solid rings of constantly increasing size, and that we so 
modify the volume of oil that the length of the meridian node, that is, the dis- 
tance from its summit to its point, shall remain the same. When the diameter 
of the solid ring is very considerable, the perpendiculars corresponding to the 
different points of the node will be all very large, so that at all these points 
1 : 
the term 35 of the equation of equilibrium will be very small, and we perceive 
that this term will tend towards zero in proportion as the diameter of the solid 
1 P 
ring tends towards infinity ; but it cannot be thus with the term w for if this 
last also tended towards zero, the liquid figure would have for a limit of its varia- 
tions the catenoid, which is evidently impossible under the conditions implied— 
that is to say, when the node is of constant length; we can always, then, im- 
agine the solid ring so large that at all points of the meridian node the term 
F 
N shall be very small relatively to the term a The latter, which expresses 
the meridian curvature, should now, in virtue of the equation of equilibrium, 
vary very little on the whole contour of the node, and consequently this will 
closely approximate to the circumference of a circle. It is clear that, in this 
case, the curvature of the ares which connect the consecutive nodes of the 
complete meridian line will also be very nearly constant, and of the same order 
1 
with that of the nodes, for the term N will be also very small on the arcs in 
question. From this we perceive that the consecutive nodes of the meridian 
line will encroach upon one another, and that hence for a certain large diameter 
of the solid ring this line will have the form partially represented at Fig. 40. 
In this figure the axis of revolution is not indicated, because it is situated at 
too great a distance. 
If we imagine the diameter of the ring still further enlarged, the meridian 
curvature will still more nearly approach uniformity ; the nodes will be more 
nearly circular and will more closely encroach on one another; finally, at the 
