1879.] Definite Integrals involving Elliptic Functions. 347 

 log {1 — dn 2 (y, h') sn 2 x\dx 



Jo 



= -i^(K-2/) 2 + Klog(?^ , )-2Klogefe7 i ;') . . . (64), 



and integrating with regard to y between the limits K' and these 

 give 



loT log { 1+ ^li sn2 4^=^ K2 -^ K ' 2+ * KK ' 10 <f ) • (65) ' 



^ipg{i-dn»^,i')tt««}db=H'^-^+packg(^(6e) > 



which are in fact Mr. Roberts's results (" Lionville," t. xii, p. 456). 

 In the " Philosophical Magazine," for December, I860,* Professor 

 Sylvester has given the values of two integrals which are equivalent 

 to (5) and (28) . The method depends upon an expansion in a series 

 and is quite different to any employed in this paper. Professor 

 Sylvester notices particularly the fact that the values of the integrals 

 in (5) and (28) should be equal in magnitude. 



§ 14. Recently in the " Mathematische Annalen" of Clebsch and 

 Neumann (t. xi, 1877, pp. 567 — 570), Dr. Enneper has obtained the 

 evaluation 



dn 2 x log dn xdx — |E log V + J (1 + ft' 2 ) K — E 



by the substitution of K— x for x and further transformations : but the 

 value of this integral had been previously found in a somewhat similar 

 manner by Mr. Roberts, in a note in " Liouville's Journal " for 1846 

 (" Extrait d'une Lettre addressee a M. Liouville," t. xi, pp. 343, 344). 

 Dr. Enneper, in his paper, by the use of (33) shows that 



| arc tan { \ dx 



Jo I dn (x + a) dn (x — a) J 



^arctan { dn (»+«0 0°-") } dx=j^K . . (67), 



and it may be observed that, by following the same method, we have 

 at once 



* " Notes to the Meditation on Poncelet's Theorem, including a Valuation of the 

 two new Definite Integrals 



* *°g cos< P 4 fe log{l+ A/rry(coB<p)^ f „ 



o yi-^Ccos^) 2 ' J o /l-a»(oos$) 2 



"Phil. Mag.," ser. iv, t. xx, pp. 525 — 533 (1860). See also, by the same author, 

 "Note on certain Definite Integrals," " Quarterly Journal of Mathematics," t. iv, 

 pp. 319— 324 (1861). 



2 B 2 



