1879.] Definite Integrals involving Elliptic Functions. 345 



necessary changes in the limits. This is evident, since 



dam x -i 

 =dn x, 



dx 



and as am K=|tt, am 2K=tt, we see that to the limits K or 2K and 

 correspond respectively the limits \ir or ar and 0. 

 Thus (5), (6), (7) maybe written 



|W loggin g ^ _ i7rK ^_i K k ^ 

 Jo (l-& 2 sin 2 a0 4 2 5 ' 



Jo (l-^sin^)^ 4 2 & 



^log(l-^sin^) K } ^ 

 Jo (l-& 2 sin 2 ^ 8 



while, for example, (57) becomes 



and the other integrals may be similarly transformed. 



In this form several of the above integrals have been obtained by 

 Mr. William Roberts in his papers " Snr l'Evaluation de quelques 

 Integrales Definies par les Fonctions Elliptiques " (" Lionville's 

 Journal," t. xi, 1846, pp. 157—173), "Snr l'Integrale Definie 



p^ lf)g(l+^sin 2 0) r ^,, , Id 4 71 _ 4 7 6 ) 5 and "Note sur quelques 

 Jo (1 — & 2 sm 2 0)* 



Integrales Transcendantes " (Id., t. xii, 1847, pp. 449 — 456), which 

 contain evaluations equivalent to (5), (6), (7), (16), (38), (44), (48), 

 (57), (59), (60). In the first of these papers the value of the integral 

 in (7) is found by means of the transformation V tan tan -^r— 1, which 

 is equivalent to the substitution of K— x for x ; and in the last paper 

 the values of the integrals in (5) and (6) are found directly from the 

 ^-products, viz. : 



s /2Ka?\_ 2qi g . n x (l-2g 2 cos23? + ff 4 )(l-2g 4 cos2ff + g 8 ) . .. 



U / # Sm X (1 - 2q cos 2x + <f) (1 - 2(f cos 2x + <f) . . . 

 cn^Wf^lV cos x a + ^c os2^+^)(l + 2^cos2^ + g B) . , , 



\ 7f J \JcJ^ (l-2 g cos2^ + 2 2 )(l-2 2 3 cos2^ + 2 6 ) . . . 



by means of the integral (1), which gives 



(V 



log(l— 2q n cos2x + q^ n )dx=0. 



The equation (38) is also obtained directly from the ^-product for 

 ; this is of course practically equivalent to the use of the cosine 

 series (52) for Jog Qx quoted in § 11. 



VOL. XXIX. 2 B 



