APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 121 



connecting the points B, and B^,, and Bj and B^ by links of equal 

 length, and assuming 



where e is any real quantity satisfying the implied condition. 



These two links always cross each other at a point Q which does 

 not change its distance from O during the motion. 



4. The Open Lhik-Work. Consider a link-work of the pre- 

 scribed kind which does not close or which is not completed so as to 

 form a closed link-work. Suppose there are m cells in the link-work, 

 and that the last cell does not overlap the first.* In this manner an 

 angle A^4.jOAj is formed between the last and first cell. This angle, 

 which will be designated by ^, is variable during the motion, and can 

 be expressed by elliptic functions, for, 



is a function of the argument u. 



The condition for a maximum or minimum of the angle <^ is 



d(f>^ da^+, _ dx^+, da, ^ ^^q /27) 



du dx^^, du dii, du 



According to previous formulas 



£^ ■■-(.-' )(.-.)' '""' |=./(--)(-*)(--o)(.-«j). 



Substituting these expressions, with the proper indices, in (27), the 

 condition reduces to 



{x-a) {p—d) = {x^j^—a') {x^^.-d), 

 or a??— «^+t=(«+^)(a^.-».+i). (2«) 



This equation is satisfied in two ways: — 



(1) when a;,=a;„+„ (29) 



(2) when x,-\-x^^,=a^d='l{K'-\-i»). (30) 



(*) This assumption is made in order to have a clearer idea of the link-work, although the 

 results hold also in the most general case. 



