122 UNIVERSITY OF COLORADO STUDIES 



In the first case the condition a?i=a?ni+i ^^^^ ^^^ assign any relation 

 between R, r, and e and holds therefore for every proper link-work. 



Considering a complete revolution of a link-work, Fig. 1, it can 

 easily be proved that there are only two positions of the link- work 

 possible where x^^x^j^^. This is the case every time that the cell 

 has a symmetrical position with regard to the axis OQ, which, in 

 these cases, bisects the open space of the link-work. Suppose now 

 that the link- work makes a complete revolution, starting from the 

 position of the maximum angle. The angle cannot pass through 

 zero, because the system would then be permanently closed, so that 

 there must be a minimum between the two maxima. Similarly there 

 must be a maximum between two minima. This result may be 

 summed up in the theorem: — 



TJie angle formed hy an open linh-work can assume only one 

 maximum and one minimum during a complete revolution. 



The maximum and minimum angles are both bisected by the 

 diameter OQ. 



If the angle becomes zero, it will remain zero. In this case we 

 still have iz?i r=ia;„,^j (coincident) ; but for every position of the link-work. 

 Thus, we see that the case of a closed link-work is included in case 

 (1). The second condition a?j-|-a?^+i =2(E,^4-/*^) can only be satis- 

 fied in a singular case, since se^-\-x^j^^, for all possible link-works, 

 with constant values of R and r, may be considered as a function of 

 m and €, having for all values of m and e a constant value. From 

 formula (16) it appears that a?i+iK„+i can be independent of nt and e 

 only if e=0. In this case a;i+'«m+i==2r'^, and, according to (30), 

 11=0. There is no proper link-work. 



Without entering into mechanical details of the link-work it is 

 interesting to mention the seemingly paradoxical fact, that all our 

 link-works have one degree of freedom in their motion, although the 

 closed link- work satisfies the condition of a rigid frame- work. 



5. Geometrical Transformation of the Link-Work, With 

 A„ Aj, A3, . . ., in the previous figures, as centres and r as a radius 

 describe a series of circles as before. These circles all pass through 

 O and intersect the circle of centre Q and radius R in the points B,, B^; 



