v] OF MINIMAL SURFACES 219 



of this latter will be a circle of infinite radius, i.e. a straight line 

 infinitely distant from the axis ; and the surface of rotation is now 

 a 'plane. If we imagine one focus of our ellipse to remain at a 

 given distance from the axis, but the other to become infinitely 

 remote, that is tantamount to saying that the ellipse becomes 

 transformed into a parabola ; and by the rolling of this curve 

 along the axis there is described a catenary (D), whose solid of 

 revolution is the catenoid. 



Lastly, but this is a little more difl&cult to imagine, we have 

 the case of the hyperbola. 



We cannot well imagine the hyperbola rolling upon a fixed 

 straight line so that its focus shall describe a continuous curve. 

 But let us suppose that the fixed line is, to begin with, asymptotic 

 to one branch of the hyperbola, and that the rolling proceed 

 until the line is now asymptotic to the other branch, that is to 

 say touching it at an infinite distance ; there will then be mathe- 

 matical continuity if we recommence rolling with this second 

 branch, and so in turn with the other, when each has run its 

 course. We shall see, on reflection, that the line traced by one 

 and the same focus will be an " elastic curve " describing a suc- 

 cession of kinks or knots (E), and the solid of revolution described 

 by this meridional line about the axis is the so-called nodoid. 



The physical transition of one of these surfaces into another 

 can be experimentally illustrated by means of soap-bubbles, or 

 better still, after the method of Plateau, by means -of a large 

 globule of oil, supported when necessary by wire rings, within a 

 fluid of specific gravity equal to its own. 



To prepare a mixture of alcohol and water of a density precisely 

 equal to that of the oil-globule is a troublesome matter, and a 

 method devised by Mr C. R. Darling is a great improvement on 

 Plateau's *. Mr Darling uses the oily liquid orthotoluidene, which 

 does not mix with water, has a beautiful and conspicuous red 

 colour, and has precisely the same density as water when both 

 are kept at a temperature of 24° C. We have therefore only to 

 run the liquid into water at this temperature in order to produce 

 beautifully spherical drops of any required size : and by adding 



* See Liquid Drops and Globules, 1914, p. 11. Robert Boyle used turpentine 

 in much the same way. For other methods see Plateau, op. cit. p. 154. 



