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



we have 



f2K 



I log (1 + cn x)dx= — jTrK'^— K log h (21), 



I log(l — dna;)<fo5=— fa-K' + KlogZ; ..... (22), 



j 2K lo g a + dna3)^= frrK' + Klogk (23), 



|^log(dna3 + cn^)^=-i 7 rK , + Klog^^ . . . (24), 



log (k't + dnx + tf cnx)dx=±7rK' + K\og (M'*) . (25). 



Since sn(2K— u) = sn u, dn(2K— u) = dnu, cn(2K — u) = — cn u, it 

 follows that 



j*2K [*K 



0(sna?)cfo=2 0(sn#)cfce 



Jo Jo 



i2K fK 

 0(dna;)^a;=2 (p(dnx)dx J> . (26), 



(*2K [*K 



0(cn^)^^= J {0(cn#) + 0( — cnx)}dx 

 and therefore from (22) and (23) we dednce that 



^log(l-dna»)(fe=-fTK^+JS:iog* . . . . (27), 



| K log (l + dn^)^= ^TrK'+lKlogZ; . . . . (28). 

 Bnt (21) only gives the valne of 



fK 



{log (l—-cn aj)+ log (1 + cn 



• Jo 



and is, therefore, equivalent to (5), while (24) and (25) are merely 

 transformations of (5) and (28). 



We also see that in (24) and (25) the integrals may be written 



(*2K 



log (dnx — cnx)dx (29), 



Jo 



J"2K 

 log (k' 2 + dnx — T£cnx)dx (30), 



Applying (26) to (17) and (18) in the forms (19) and (20), we see 

 that 



j^ilog(cn^±sn^dn^) 2 ^=-i7rK / + Klog^^. . . (31), 

 | 2K log(dn^ + ^sn^cn^)^=--j7rK / + Klog^^. . . (32). 



