Prof. Tait on Listing's Topologie. 37 



To find the divergence, take the case of /* / v greater than 

 m J n; and a, so found, nearer to A than to A x . Draw a c per- 

 pendicular to AA b and let the spirals through a, parallel to 

 B A and B A 1 respectively, cut A A x in d and e. Then the 

 divergence is 27rAc/ AA X . This is obviously greater than 

 2irAd/ AA X (i. e. 2mv / n), and less than 2irAe/AA 1 (i. e. 

 2irfji I m)\ and can be altered by shearing the diagram parallel 

 to AA 1? or (what comes to the same thing) twisting the stem 

 or cone. To find its exact value, draw through B a line per- 

 pendicular to AAx (L e. parallel to the axis of the stem or cone), 

 and let C, the first leaf or scale it meets, be reached from B by 

 r steps along B A, followed by s steps parallel to B A 1# Then 

 the divergence is easily seen to be 27r(/jbs + vr) /(ms + nr); 

 and we have the complete description of the object, so far as 

 our science goes. 



In the figure, which is taken from an ordinary cone of 

 Pinus pinaster ^ we have m = 5, ft =8; whence yu,=2, v=3. 

 Also r = 3, s = 2; and the fundamental spiral {perverted) is 

 therefore right-handed, with divergence 27rl3/34. 



Should m and n have a common divisor p, it is easily seen 

 that the leaves are arranged in whorls i and, instead of one 

 fundamental spiral, there is a group of p such spirals, forming 

 a multiple-threaded screw. Each is to be treated by a process 

 similar to that above. 



(8) The last statement hints at a subject treated by Listing, 

 which he calls paradromic ivinding. Some of his results are 

 very curious and instructive. 



Take a long narrow tape or strip of paper. Give it any 

 number, m, of half-twists, then bend it round and paste its 

 ends together. 



If m be zero, or any other even number, the two-sided sur- 

 face thus formed has two edges, which are paradromic. If 

 the strip be now slit up midway between the edges, it will be 

 split into two. These have each m/2 full twists, like the ori- 

 ginal, and (except when there is no twist, when of course the 

 two can be separated) are m/2 times li?iked together. 



But if m be odd, there is but one surface and one edge ; so 

 that we may draw a line on the paper from any point of the 

 original front of the strip to any point of the back, without- 

 crossing the edge. Hence, when the strip is slit up midway, it 

 remains one, but with m full twists, and (if m> 1) it is knotted. 

 It becomes, in fact, as its single edge was before slitting, a 

 paradromic knot, a double clear coil with m crossings. 



[This simple result of Listing's was the sole basis of an 

 elaborate pamphlet w r hich a few years ago had an extensive 

 sale in Vienna; its object being to show how to perform (without 



