84 Mr Greenhill, On Conjugate Functions [Mar. 21, 



7. The integral 



[ dz 



] ^(l-z\ 1-fcV) U ' 

 saving z — sn w, 



may however be written 



/ - - '"* -= fa " 



lered as a part 



f i 



J \]{z — a.z — b.z — c.z — d) 



and may be considered as a particular case of 



dz 



where 



a— t, 6 = 1, c = — 1, tZ = — t» 



and therefore 



_1 



l 



* + & 2 



— =r^ cn2 ^ 1+ *>> 



and if \ denote the new modulus, 



2 

 , 2 ' fc 4A; 



F^f (1+A)!: 



equivalent to Landen's transformation. 



Therefore, putting £ (1 + &)w = ff' + 177', 



r 2 



J = j-^ sn (f + ^') sn (f - M7 ), 



r 2 



^ = — _ k en (f ' + z't/) en (| ' - m/), 



r„ 1 + & 



_s 



j— ^ dn (I' + m/) dn (f ' - m»') ; 



