Lazytongs and Lattice-icorks. 381 



take place either in the n direction or in the m direction, or 



in the - direction or in the — direction. 

 n m 



But the bars AB &c. which lie in what would then be 

 called a bishop's move, would also have their extremities in 

 an equiangular spiral haying the same pole as the other sets 

 o£ spirals. The angle between successive chords (external) 



will be in this case a — /3 + y~ 8, and the change ratio of the 

 chords mn, so that the angle characteristic of the spiral will 

 have for its tangent 



^fi + y^S 

 logjnn 



Circles and straight lines are only limiting cases of equi- 

 angular spirals. They may therefore be awaited among the 

 cases arising from chanffins: m. n or the anode between the 

 cross-bars. 



If n = l the n spirals become circles, and since in that case 

 7 = 8, the m spirals become straight lines. 



If m = l the m spirals become circles and the n spirals 

 straight lines. 



If both in and n are equal to unity, both systems are 

 straight lines. 



If 7rm=l, in which case AB is bisected in E, there is no 

 change of scale along the bishop's move in the direction AB. 

 In consequence the spirals through AB are circles. 



Similarly if m = n CD is bisected, and such lines lie on 

 circles. 



Such lines as AB, BK, &c. may under some circumstances 

 lie in a straight line. 



The displacement in angle of such lines is (a — (3 + <y— 8). 



If this is equal to zero 



« + 7=/3 + S 



Thus the circumscribing circle must have AB for a 

 diameter. Therefore as D and C are both on that circle 

 CD must either be equal to AB, in which case the two chords 

 bisect each other (m=w=l), or CD is less than AB. It is 

 therefore only the longer of the cross-bars which, by the 

 variation of the angle between them, can come into a straight 

 line. The longer cross-bar is also that one which is divided 

 most unequally, since the product of the segments is the same 

 in the two bars. 



Phil. Mag. S. 6. Vol. 14. No. 81. Sept. 1907. 2 D 



