198 M. G. Van cler Mensbrugghe on the 



more marked the more strongly curved the part in which the aper- 

 ture is situated. The question suggested itself, if the film does 

 not acquire a greater area in proportion as it is more profoundly 

 altered, and hence does not oppose an increasing resistance to this 

 augmented surface in proportion as this further removes the figure 

 from the form of original stable equilibrium. This supposition 

 has been justified by the following experiment. On the helicoidal 

 film a cocoon-thread, 50 to 60 millims. in length, is arranged, 

 the two ends of which have been tied; after having burst the 

 film interior to this contour, the thread, by the effect of its 

 weight, may be moved to all parts of the surface except near the 

 axis of the helicoid : it may, indeed, be brought by force to this 

 part of the film; but then as soon as ft is free it moves from the 

 axis, spite of its weight, and regains the portions with weak cur- 

 vatures. 



3. Caienoid. 

 If, as a third hypothesis, we put 



<j>(u) = \/l-us, >^(v) = \/l-v 2 , 



equations (1) represent, as we know, a catenoid, and the asym- 

 ptotic lines are given by the following relation, where r, co repre- 

 sent the polar coordinates, and a an arbitrary constant, 



But by a general property demonstrated by Mr. Roberts, these 

 lines cut the meridian curves of. the catenoid at an angle of 

 45°*. Thus they are helicoidal curves, the spires of which 

 are larger the further they are from the circle of the gorge; 

 they cannot, therefore, give in all their points the same radius of 



* See the memoir by Mr. Roberts, p. 312. By this property the above 

 relation is directly obtained ; for, on the one hand, the meridian catenary 

 is represented by the relation 



e 



on the other hand the equation of the trajectory forming at each point an 

 angle of 45° with the meridian line is given by the formula (see the memoir 

 of the Abbe Aoust, " On the Trajectories which cut the Meridian Curves 

 of Surfaces of Revolution at a constant Angle," Liouville's Journal, vol. xi. 

 p. 184) 



whence 



dz A , (dr\ 2 dz 



-\-a, and r= - (e w< 



