Idea of Vonceie^s Theorem* 313 



not, nor ever Can be the best and closest of their respective kinds*. 

 To fix the ideas, let us confine ourself to the second Ponceletic 

 approximation to Va + bx + cat 1 *, viz. that which has the form 



~-r 1 where \, fju 3 v are to be determined. The problem 



to be solved is the following. 

 Let \-f-^# + v# 2 =V, 



(l+qx) \/a + ba: + cx 2 = U ; 

 it is required to assign the four constants X, ft, v, q, so that the 



limit to the integral, viz. 



(p+q+ ^pt+q'f+ip+q- ^p+q 9 f Vp 2 +? 2 

 or 



(p+q+Vp*+q*) i -i-(p+q- A /p 2 +q 2 ) i l 



in which formulae the greater i is taken the closer will be the approxima- 

 tion. I am not aware that any of these limits to F(c) (even the simplest of 

 which, viz. those given above, may have some value for computational pur- 

 poses, and have fallen thus very incidentally in my way) have ever before 

 been noticed. 



It is not unworthy of notice that the second superior limit to / -= . 



viz. 7-— — £*?ii;i is an arithmetic mean between the first superior and 



(p + q)(p a +q i y ... 



first inferior limits, and consequently our second superior limit to the 



integral when b is indefinitely small becomes log t-Hj which brings the 



constant much nearer to its true value than did the use of the first limit ; 

 and as this approximation will evidently not stop at the second step of the 

 process, we may safely infer that the integral derived from either formula 

 when i=Mco (for all values of b, whether finite or indefinitely small), not 

 merely bears to F(c) a ratio differing infinitely little from that of equality, 

 but is absolutely equal to, and may for all analytical purposes be employed 

 to represent F(c). 



I have been at the trouble of calculating the inferior limit afforded by 

 the second approximation, and find that for b indefinitely small it is 



9 7T 1 



log- -f =rjl3> i. e. log ^ + 1*2977 ; the superior limit has beenshown to be 



1 i 



log£ + 1*4785, the mean is therefore log^ + 1'3881, differing by only '0018 



from the true value ! As the constant continues for all values of i to be a 

 multiple of 7T, the ith. approximations a supra and ab infra, which are 

 always effectible, will give (on making i=so ) two new expansions for 7r,one 

 infinitesimally in excess, the other infinitesimally in defect of its true 

 value expressed as a multiple of log 2, which it might well repay the trouble 

 of some young analyst to develope. 



"**' That the fractional forms derived from the linear substitutive form are 

 not the best of their respective kinds, appears immediately, so far as the 

 Phil. Mag. S. 4. Vol. 20. No. 133. Oct. 1860. Y 



