344 KEPORT — 1873. 



whence we have 



which therefore constitute the solution of equation B, 

 Hence also the solution of the equations 



IS 





where r^^ ^, r^^ ^, Sj_ j, s^^ ^ are any whole numbers. This formula then con- 

 tains the required solution ; and therefore, substituting for iv, in the equa- 

 tions connecting u and w at the end of section (1), we have 



We have already stated that this transformation corresponds to the periodic 

 system 



al (u^ + 2K^^ ^, M^+2K, J^=a?(tf,w^)2 when a = l or 3. 

 In the same way, if we take the periodic system 



a?(Mj-|-2K, .J, ^l3+2K^^^)l=al(u^uy^ when a = l or 3, 

 we have 



(m' -n')(yK,. , + 2K,^ ,)=C,, ,0',, ,-Hs,, ,r,, , + s,, ,r,, ,) + C,, ,(r,, .^ + s,^ ^r,^ , + s,, ,r,, J. 

 We shall also have, if Ave take the periodic system 



al(u^ + 2iK\^ J, u.^ + 2iK'.^^ i)!=«?(WiW^)!. where a=l or 3, 



Moreover, taking the system 



alitc^ + 2{K\^ .. «,+2iK',, ,)^„=aZK, .u^, 

 we shall have 



Now we have already proved that 



h I 



- HK,,K-,,-K,.K',.) 



