114 UNIVERSITY OF COLORADO STUDIES 



cipal cases may occur: (1) the link- work will close after a certain 

 number of additions of cells, i. e., the last point A obtained in the 

 construction will coincide with the first of the points A; (2) the 

 link-work does not close. 



To discuss the conditions of a closed link- work assume that there 

 are n cells in it, so that the point K^j^^ of the nth. cell OA„B„A„+, . 

 QB„ will coincide with the first point Aj. The argument belonging 

 to the angle a, or the point A, being ?/, the argument of A,^ will be 

 u-^-h, of A3?/+2/^, . . . , of A„+,^/+w//. But A„+, coincides with A„ 

 hence, designating the periods of the elliptic function X(w) by ^w 



and 2i^, 



u-{-7ih^n (mod 2w, 2i«j). 



This condition is satisfied if 



h = (mod—, — ), 



71 11 



2m2w-{-27nj2wj , ^ ^ , 



or h= , (11) 



71 ^ ^ 



where m, and m^ designates integers. Consequently the problem of 

 a closed link -work is solved if h is given one of the values contained 

 in (11). This condition necessarily requires a special arrangement 

 of the link- work; but it does not assign any particular value to the 

 argument u. Thus, the first point A, of the link-work may be 

 chosen anywhere on the circle having O as a centre and OAj as a 

 radius; the link- work closes every time and contains n cells. This 

 result may be stated in the theorem: — 



Jf a link-work of the jprescirihed kind, lased upo7i two fixed 

 circles [centres and Q, radii OA^ and QB^) closes and co7itains 

 n cells, then every other link-work hased upon the same tioo circles 

 closes and contains n cells. 



It is clear that the fundamental relation (3) also holds in the 

 cases of limited and unlimited link-works. The previous theorem, 

 however, only holds for a closed link-work. 



Sll. Geometrical Transformation of the Link- Work. 



In Fig. 5, with A,, A^, A3, . . . as centres and r^ as a radius 



