1879.] Definite Integrals involving Elliptic Functions. 333 

 whence the products (2), (3), (4), give 



j K logsn^=-i7rK , -JKlog^ .... (5), 



j^logcna^ aJ =-i 5 rK'+iKlog^. . . . (6), 



log &nxdx=j;Klogk' (7), 



o 



which are the analogues in elliptic functions of the integrals 



I log sin«c?a3=|7rlog (|) (8), 



Jo 



logcosa5&u=j7rlog (£) (9). 







10 



The other formulas, such as, ex. gr., 



£ 



snwsn 4w} . . . sn {u-\-4(n— l)w} = ^Lj Sn (^|' 

 do not lead to new integrals, for this product gives 



f4K 



log sn (u + x) dx = — 7rK' — 2K log fr, 



Jo 



in which the sign of sn (u-\-x) when negative is to be changed, so that 

 the quantity under the integral sign should be written J log sn 3 (w-f a?)- 

 This result, however, may be readily deduced from (5). 



The remaining formulae on page 51 of the " Fundamenta Nova " 

 only produce the equation 



I 2 l°o fl— — — \ dx= I log (l—& 2 sn 2 w sn 2 #)(&», 

 Jo V snw Jo 



which maybe otherwise deduced from the formulas (44) and (45) of 

 §10. 



§ 4. In the second real transformation 



n 



so that we find 



j Jlogsn^da;=— JK'logife ....... (10), 



^ K \ogcnixdx=i7rK+iK'log(^j .... (11), 



logdn^=|7rK + iK'logZ;' (12), 



