APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 113 



Sin B _ l/-[pf-(R+r3)-][pf-(R-nr] 

 2R;' 



SO that 

 da, 4Rr,p, • dp. 



sin ^. y^p. - (^,+,)^] [pi _ (r- eY] [p\ - (R+r,)'^] [pf - (R - r,)^] 



To abbreviate let (R+r)'=a, (r+e)'=&, (r— e)'=c,(R — r)'=r/, 

 p\—i\ p, • dpi=^ d'X, 80 that finally 



sin /3, ^/[x—a) {x — h) («—<?) (« — ^) 

 In a similar manner, if QK^=p^^ and pl=y, 

 da^ SR^j • <fy 



(5) 



(6) 



Puttincr 



sin ^3 V{y-a) {y~b) {y-o) {y-d) 



r ^" (7) 



Jc y^{x-a) {x-h) {x~c) {x-d) ' ^ ^ 



dy 



Je |/ 



(8) 



\/{y-a) {y-b) {y-c) {y-d) 



according to equation (4) we have: — 



v—u=^h (constant). (9) 



By inversion of the elliptic integrals (7) and (8) the elliptic func- 

 tions 



x = \{n), y^Hv) (10) 



are obtained. In this manner the cosines of the angles a, and a.^ 

 may be rationally expressed by elliptic functions, and it is found 

 that the difference of the arguments belonging to these angles is 

 consta/nt and independent of the position of the cell. 



2. As indicated in Fig. 5, other equal cells (OA2B2A.J • BjQ), 

 (OA3B3A^ • B3Q) . . . may be added to the first, which together 

 form a general link- work. In this process of adding cells two prin- 



