APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 119 



Substituting in this expression the values of k and q in terms of R, 

 r, e, it is easily found that condition (23) reduces to 



Il=r. (24) 



Thus, the three cell link-work is completely determined by fixing r 

 and c. In Fig. 7, OQ=e and OAj=r. Now E,=r, hence, in this 

 case, A,Bi = QBi. Having fixed the point Bj it is an easy matter to 

 complete the construction of 



A2 Bi 



Fig. 7. 



the cells AjBiAoO, OAoBoAj, OAsB^Ai. It is seen that QBi^QB.^ 

 QB4, so that also QB2A2B1, QB1A1B4, QB4A3B2 may be considered as 

 cells of the link-work. The points O, A,, A2, A3 may be interchanged 

 with the points Q, Bi, B2, B3 without changing the character of the 

 link-work. 



3. Example of 4 Cells. In this case the value of y as given 

 by formula (17) is also equal to the value of x in formula (20), i.e., 



^ [c-d)+{b-d) vw' 



Substituting in this equation for a, b, c, d, jp, and k' their values in 

 terms of R, r, e, the conditions between E, r, and e is found: — 



2r'=R'+e\ (25) 



