Examples of Groups 



17 



We pass to a complementary modulus 

 by s?i 2 (zw, k')=- -sc 2 (w,k) [analogous to sm' 2 6= — ta?i 2 hi] 



whence f\ ( iw, k' ) =f\ (w).(/3&); 

 by dc-(iw, &')=d?i 2 w.(fi8) 



whence fi(iw, k')=fi(w, k),(ft8) 

 and fz(iw, k')—fz(w t k) . (/3o) ; 



finally by csc' 2 9dr(iw, k')=sec i 6dc i (^7v ) k)\ 

 whence f±(iw, k')=fi(zi>, k).(fi$). 



So that again the subscripts can be dropped and we can write 



f(izv,k')=f(w,/e).(J38) 



or / (iw, cos6)=/(w, sin0).(/38). 



But the operations of passing to reciprocal and to complementary 

 modulus generate a group. In writing the scheme of the group, 

 only the arguments and the moduli are put at the corners of the 

 hexagon. The dotted lines indicate (ySy), the full lines (/38). 



*>*s<n&. - y ft* cost 



W. 



<t*j*e 



vrzs*d,see& 



^oJ&inOjtoiQ 



247 



