1879.] Definite Integrals involving Elliptic Functions. 337 

 § 8. Since 



^ K (j)(%)clx=^(j)(K-x)dx (33), 



it follows from (27) and (28) that 



K log(dn^-^)^=-| 7 rK / +iKlog(^') . . . (34), 

 o 



^log(o\nx + Jc')dx = J«-K , +iKlog(Jfeft') . . . (35). 



It may be remarked that the transformation (33) applied to the 

 functions sn, cn, dn does not snffice to give the values of the integrals 

 in (5) and (6), although we thus immediately obtain (7) ; for 



I log dn xdx = I ' log dn (K — x) dx— [ log h 'dx — \ log dn xdx, 

 Jo Jo Jo Jo 



f K 



giving log dn xdx = |K log h' ; 



Jo 



fK fK fK 



but log snxdx = \og cnxdx — log duo. xdx, 



Jo Jo Jo 



|*K fK fK |*K 



and log cnxdx = \og~k'dx-\r\ log sn xdx — log dn xdx, 



Jo Jo Jo Jo 



only lead to one equation between the integrals of log sn x and log cn x, 

 viz., 



f K f K 



log cn x dx — log sn x dx = |K log h f . 



Jo Jo 

 § 9. Putting x=y in the formula 



e(x + y)Q(x-y) = ^^a-^^^^y) • • - (36), 



it becomes 



whence 



e(2a)=|J(l-^sn^) ..... (37), 



K log eOdx + I log (1 — W sn 4 x)dx. 



o jo 



| loge(2^)^=4| K loge^-3 



fK fK 



Now log 0(2,7?) dx=\ logQxdx, 



Jo Jo 



so that the equation is 



j log 0^=Klog 00— l| K log (1 — ~k 2 sn*x)dx, 

 and substituting the value of the last-written integral from (16) and 



