v] OF FIGURES OF EQUILIBRIUM 373 



The nodoid is, like the iinduloid, a continuous curve which keeps 

 altering its curvature as it alters its distance from the axis; but in 

 this case the resultant pressure inwards is negative instead of 

 positive. But this curve is a complicated one, and its full mathe- 

 matical treatment is too hard for us. 



In one of Plateau's experiments, a bubble of oil (protected from 

 gravity by a fluid of equal density to its own) is balanced between 

 annuh; and by adjusting the distance apart of these, it may be 

 brought to assume the form of Fig. 106, that is to say, of a cyhndet 

 with spherical ends; there is then everywhere a pressure inwards 

 on the fluid contents of the bubble a pressure due to the convexity 



Fig. 106. 



Fig. 107. 



of the surface film. This cylinder may be converted into an undu- 

 loid, either by drawing the rings farther apart or by abstracting 

 some of the oil, imtil at length rupture ensues, and the cylinder 

 breaks up into two spherical drops. Or again, if the surrounding 

 liquid be made ever so little heavier or hghter than that which 

 constitutes the drop, then gravity comes into play, the conditions 

 of equihbrium are modified accordingly, and the cylinder becomes 

 part of an unduloid, with its dilated portion above or below as the 

 case may be (Fig. 107). 



In all cases the unduloid, like the original cyhnder, is capped 

 by spherical ends, the sign and the consequence of a positive 

 pressure produced by the curved walls of the unduloid. But if 

 our initial cyhnder, instead of being tall, be a flat or dumpy one 



a small funnel in a large one, wet them with soap-solution, and draw them apart; 

 the film which develops between them is a catenoid surface, set perpendicularly 

 to the two funnels. On this and other geometrical illustrations of the fact that 

 a soap-film sets itself at right angles to a solid boundary, see an elegant paper by 

 Mary E. Sinclair, in Annals of Mathematics, viii, 1907. 



