Meditation on Poncelet's Theorem, 529 



where 



r_C T ja COS0 



Jo (l + * 2 )-(c 2 +tf 2 )(sin0) 2 



= W+WW W 1 *** ^^)~^W}' 



Let x — tan cj>, b = a/ 1 — c 2 ; 

 then 



F ( c )-irvi-^cos^{ i ° g ( (scc ^ + ^^^ : ^- io s & } 



-|logi.F(4)4§R, 



where 



ir 



J* Y l\/l-bHcos6) 2 J 



f¥ log(l + \/l-6 2 (cos</>) 2 ) 

 1 #d> ^ — ^-. 



Jo ' */l-Z> 2 (cos</>) 2 



It will presently appear that these two definite integrals are 

 equal to one another ! 



f° 

 Let V 2r = 1 (cos cf>) 2r log (cos </>). 



Then we may easily establish the formula of reduction, 

 _ 2r-l v 1.3.5..(2r-3) tt 



2r ~ 2r V2r - 2 2.4<6..(2r-2)2r2 j 



7T 



and since (as is well known) V =— log 2, we have 



V4= l5f( lo s 3 -r3-3Ti)' 



„ 1.3.5 tt/. 1 1 1 \ 



Ve= 2A6 2 ( l0 S 3 -r2-0-0j' 

 &c. = &c. 



