1881.] certain Definite Integrals. 453 



Jo I e-t>~>-cc <r«i—<* e-K—a e^—a) 



f 2 — (/ * -/ r i )=*v(fJ_-/0*l . (204). 



J tan#l p— *fccotan0 £> + £Cotan#J l"p + l J 



ri^(/_L--/-^-l=^(/J_-/oj .... (205). 



rt{/i-/J r U{/L/0} (206). 



Formula (199) leads to some results, which may well be noticed 

 separately. 



T de ■ 6 - " J - f -l 1 



Jo a + b sine 2^1""^^ Stf~&f * ' (207) ' 



This may be written (for a particular case) : — 



f* d ±l __^tan-i_^-j . . . (208). 



Jol + 2asin0 + ^ 2(a2-l)l a?-l cfi-l J v J 



Jo sin^+ ... esin ?i _ 2 J ... + e sin^fl 



(209). 



ede 



Hence 



J7r cat? 

 oVr+^i^ (210) ' 



can be expressed as an elliptic function. 



The integral J ^ ai ^ a £ = ^ (stated to be due to Cauchy) which is 

 proved in Gregory's examples, gives us 



f°°^J 1 cotan|— eotan4=-(l-- > ) • ■ (211). 

 Jo 6 I 2 n 2 W J 2\ 2V V y 



Let (%) increase without limit, then this integral becomes — 



r^/i-cotanflj^ (212). 



Jo e 16 J 2 v 



