For computing the E{u,k) column in our tables we used also 



E(u + v, k) =E{u, k) +E{v, k) -KHn{u, K)sn{v, K)sn{u + v, k) 

 For tabulating sn(«,/c), cn(w, k), dn(w, k), E{ii.k), «/) = sin-i(sn(w, k)) 

 we set M= -^ K = r°, whence 7^/= -^ • — =r° for sin, cos, sinh, cosh. 



Also K =sin 6° 



/c' = cos e° 



Again, we write 



.. Q. _ sinr°-g^sin3r° + g'sin5r°-g"sin7r° + g''°sin9r° . . . . 

 ^'' ^~ l+<?2 + g« + ^i2 + g^o+ .... 



B(r°)= A(90°-r°) 



, o l-2gcos2r°+2g^cos4r°-2g^cos6r°+2g^''cos8r°- . . . . 

 ^^ ^~ l-2(7 + 2(?*-2g» + 2g^«- .... 



C(r°)= D(90°-r°) 



whence 



sn(r°.^°)=D(90°)-^|^ 



dn(r°,r)= ^ ^^''°^ 



AlsoZ(r°,r)=^ 

 K. 



D(90°) D(r°) 

 q- sin 2r ° — 2^* sin 4r ° + 3g^ sin 6r* 

 1 - 2^ cos 2r° + 2g* cos 4r ° - 2^^ cos 6r ' 



- V« sin 8r° + 5g-^ sin 10r° - 



+ 2qi«cos8r°-2^2^cosl0r°+ . . . 

 whence 



Beginning with r° = V we obtain succeeding values £(2°), E{Z°), etc., 

 bymeansof £(r° + l°)=£(r°)+£(l°)-/c2sn(r°)-sn(r)-sn(r° + r). 

 The ^ column was computed from the relation 



sin ^ = sn M 



that is, <^(r°, ^°) =sin-Hsn(r°, 6°)) 



26i 



