118 UNIVERSITY OF COLORADO STUDIES 



an (2:+2K):=— sn z, 

 or 8nX2+2K)=sn% 



i. e., 2K is the real period of &n^z. For 2=0, 8n'^2=0 and x= 

 {r—e)\ For 5 = K, &\ih=l and «= J= (r+e)'. For 2=2K, 

 sn'^^r^O and ii:'=6'=(r — ey. To find the corresponding value of a;, 

 belonging to z=K\2, we make use of the formula': — 



8n^=— =i= (19) 



where 



^^^_ i^a.-h)[c-d) 



{a-c) {b-d) 



is the complementary modulus. Thus, for s = K|2, from formula 



(16) we obtain 



J{c-d)+c{h-d)Vjr 



^- ^^_d)^{h-d)VV' ^ ^ 



2. Example of 3 Cells. As the period of sn^2 is 2K,we have 

 to put s=2K, in order to obtain the relation of R, r, e, in this par- 

 ticular closed link- work. Designating sn z\S simply by S, we have: 



sn .= 3S-4(l+^)S--f6;L-S--^'S- ^^. 



1 -6/fcS'+4(l-f/i:)^S«-3^V 



and since sn 2K^=0, the condition becomes 



^•2S«-6^'S*+4(l+^)S^_3=0. (22) 



According to formulas (17) and (15): — 



^2_ {x—c){h—d) 

 (^x-d){l>-e)' 



Designating this expression by q, the required condition is 



kY-6kq'+4.{l+k)g-S=0. (23) 



(1) For the formulas used and developed here and in the next two sections we refer to 

 Greenhill, loc. cit., pp. 120-121. 



