348 Mr. J. W. L. Glaisher on [Nov. 20, 



arc tan I — — \dx= arc tan I \dx 



Jo \ h k / Jo \dn x) 



{ arc tan (^) + arc tan (S } * 



=ij^iwt&J=|iirK (68). 



Also, n being any quantity, we see from (33) that 

 f K dx _ f K da; 



dn w a? 



= i FY V* + dn» s V 

 2 J o W*» + dn M x h' kn + dn w a;/ 



=iK. 



so that 



' K dx _! K 



o/j^' + dirM; 



, K 7 / n ? dx—^K (70), 



and, similarly, 



dx 



o^+dn re (x—a) dn w (x + a) 



>K dn w (a 5 — a) dn w (x + a) ^ 

 I o &'»+dn w (a;— a) dn M (a? + a) 



=i* (71). 



Further, if be an uneven function and w any quantity, 



whence 



=i » log & j Q K log dn ^ 



and of course there are other formulas involving and 

 dn(a? — q)dn(a; + ft) wticll may ^ e prove d in a similar manner. 



