The following two examples will be used to illustrate this method, first, 

 because the results can be readily checked by reference to our tables, and, 

 secondly, because the fundamental values q, q', K, K' are available without 

 preliminary computation. 



Example (1) : Compute sn(44°, 89°). 



For ^=89° we have 



(?= 0.40330, 93063, 38377, 8 



g' = 0.00001, 90395, 55386, 95351 

 A:= 5.43490, 98296, 25564 

 i^'= 1.57091, 59581, 27243 



44 



Also U= j:rpr K. 



Using equation (E) above, we have 



[sinh (44° ^) -q'^ sinh 3 (44° f,) +etc.' 



Gn(ir,S9°)= '^^/ ^^ ^ ^^ ' 



^°^^'^ cosh (44° ^\ +g'2 cosh 3 I' 44° ^,) +etc. 



To explain the argument (44° ^,1 we note 



^~ 90 " IK' ' ^^^"^^ '^~"90"2"^~ ~K' 



This change from the rational argument 44° on the left side of the equation 

 to the irrational argument 44° -rpj on the right side is unfortunate for the 

 computer, but seems to be unavoidable. 



Example (2) : Compute sn(88% 89°) 



2 

 Here setting u=-^K, and using equation (F) above, we have 



sn(88°, 89°)= ^'^^'^ 



Goiq') 



Using this method the authors were able to compute sn(89° 59' 59", 

 89° 59' 59") with no extravagant amount of computing. 

 For computing E{u,k) we have 



£ 



E(u,k)=Z(u,k)+ -t? -u 

 K. 



where 



„, \ _ ^'^ r ^ sin 27rt/— 2g* sin 47rt:/+3<7® sin 67rz; 



K \_\— 2q cos 2-7TV+2q^ cos 4TTV — 2q^ cos Gitv 



— 4^^® sin87rt^ + 5q'-^ sin IOttt;— . . . 



in which v— ^„ . 



4-2q^^ cos ^■rrv—2q-^ cos \Qttv-\-. . . .. 

 u 



362 



