Meditation on Poncelet's Theorem, 531 



Thus 



K =log2, 



K4= l4( log2 ^r 1 2""A)' 



&c. = &c, 

 and consequently 



f ; » iog(i+vT-y(co»»)« 



J„ ^ l-b" 2 (cos <j>f 



{lo g3+ Q) 2 (log2- I l- 3 )^ + (^)(lo S 2- i L_ 3 -L)^ +& e.}. 



Thus, then, we obtain the following remarkable equalities : 



(cos <p) 



= log h(b) + 2 F ^VWl-b*^*)* ) 



6 Jo v/l-6 2 (coscf>)2 



or 



J 



(C0S</>)' 

 log(l+\/l— 6 2 (COS(^) 2 ) 



o ^l -a* (cos £)* 



log COS (/> 



2 



7T 



= |F c +i-log4FJ. 



When 6 is indefinitely small, it is obvious from either of these 

 equations that 



F(e) = -|logS| + 21og2 = logj, 



Legendre's well-known formula previously referred to. 



The equality of the first two definite integrals in the sorites 



