Idea 0/ Poncelet's Theorem. 311 



The method, however, of Mr. Merrifield in working out this 

 conception is, I believe, entirely different from that here indi- 

 cated: how the many mathematicians of a practical stamp, 

 English and foreign, who have worked with Poncelet's method 

 during the last quarter of a century, should have managed to 



to furnish an example of this. Let 



frr- l = V(l-a*)+W 



V(l-«*)(l-c**») Vl-# 2 (l-cV) 

 (sothat6 2 =l-c 2 ), and let 



Vl-a? 2 (1-cV) ' 

 if we make 



f— z 



s/2-\ 



1 + V2 1 + vV V2 + 1 



it follows from Poncelet's theorem, that for all values of x intermediate 

 between and 1, yjsx will differ from (£>x by less than e(px. 

 Now it will easily be found by ordinary integration that 



b 



tan -1 

 c 



Hence I dx <f>x must be always less than 

 Jo 



/ . 1+c g b 



2(13^ lo ^T^ + (T3^tan-i -, 



I , 1 + e 1 6 



*•«• ^^^l^+c^ 11 " 1 ? 



when c becomes indefinitely near to unity ; that is, when b becomes inde- 



2 7T 



finitely small, this approaches indefinitely near to logr+o- But we 



know, by a theorem of Legendre, that the approximate value for the inte- 



• 4 f i 



gral in such case is log t ; so that the superior limiting value of I dx (fix, 



Jo 

 found by the application of Poncelet's method, approaches in this instance 

 indefinitely near to the value itself. The explanation of this is easy. As 

 c approximates to unity, the only important values of x in the integral 



dx 



I 



Va-^Xl-eV) 



are those which lie in the immediate vicinity of 1 ; and for all such values 

 the relative error is at a negative maximum. 



It is not a little remarkable that so rude an application of Poncelet's 

 method should serve to indicate almost with the force of rigorous demon- 

 stration the approximate formula F(c) = log r-f constant, when c ap- 



