38 Prof, lait on Listing's Topologie. 



the usual conjuror's or spiritualist's deception) the celebrated 

 trick of tying a knot on an endless cord.] 



The study of the one-sided autotomic surface which is gene- 

 rated by increasing indefinitely the breadth of the paper band, 

 in cases where m is odd, is highly interesting and instructive. 

 But we must get on. 



(9) I may merely mention, in passing, as instances of our 

 subject, the wdiole question of the Integral Curvature of a 

 closed plane curve; with allied questions such as "In an 

 assigned walk through the streets of Edinburgh, how often 

 has one rotated relatively to some prominent object, such as 

 St. Giles' (supposed within the path) or Arthur's Seat (sup- 

 posed external to it)?" We may vary the question by sup- 

 posing that he walks so as always to turn his face to a 

 particular object, and then enquire how often he has turned 

 about his own axis. But here we tread on Jellinger Symonds' 

 ground, the non-rotation of the moon about her axis ! 



But the subject of the area of an autotomic plane curve is 

 interesting. It is one of Listing's examples. De Morgan, 

 W. Thomson, and others in this country have also developed 

 it as a supposed new subject. But its main principles (as 

 Muir has shown in Phil. Mag. June 1873) were given by 

 Meister 113 years ago. It is now so well known that I need 

 not dilate upon it. 



(10) A curious problem, which my colleague Chrystal re- 

 cently mentioned to me, appears to be capable of adaptation 

 as a good example of our subject. It was to this effect: — 



Draw the circle of least area which includes four given points 

 in one plane. 



In this form it is a question of ordinary geometry. But we 

 may modify it as follows: — 



Given three points in a plane; divide the wliole surface into 

 regions such that wherever in any one of those regions a fourth 

 point be chosen, the rule for constructing the least circle surround- 

 ing the four shall be the same. 



There are two distinct cases (with a transition case which 

 may be referred to either), according as the given points A, 

 B, C (suppose) form an acute- or an obtuse-angled triangle. 



(a) When ABC is acute-angled (fig. 2). Draw from the 

 ends of each side perpendiculars towards the quarter where the 

 triangle lies, and produce each of them indefinitely from the 

 point in which it again intersects the circumscribing circle 



The circle A B is itself the required one, so long as D 

 (the fourth point) lies within it. 



If D lie between perpendiculars drawn (as above) from the 

 ends of a side, as A B, then ABD is the required circle. 



