Meditation on Poncelet's Theorem. 527 



kir 

 cos — 7- 

 , 2 7T , 2 v ^=i r . » . / • ^Y 



lo ^ + ^ + 7^47 A77'."l^v cos v n T;- 



V 1 + ( sm T) 



When i=oo these limits of course come together, and the finite 

 sums resolve themselves into the definite integral 



~ 1 ®t */i , / • == ri cos" J sin t) , 

 7tJ Vl + (sinT) 2 v ; ' 



of which, therefore, the value must be log 2. Hence, writing 

 (sin r) 2 =cos 20, we obtain 



I 



\ flsinfl log2 7r 

 \/cosT = V2 4' 



iVote C/ 



It may be shown that any of the expressions for N* derived 

 from making « = oo in the general formulae given in Note A, are 

 in fact tantamount to its representation as a definite integral of 

 a very simple kind. I shall not go into the proof of this here ; 

 it may be sufficient to indicate that it depends upon the fact 

 that the equation of infinite degree ($#)*+ (^<r) i + (^) f 4-&c, 

 may be resolved into sets of factors of a known form. In 

 the question before us, the function to be so resolved is the 

 denominator of any one of the quantities analogous to U or V 

 in Note A ; and cj)X, tyx, $x . . . become linear functions of x 

 with imaginary coefficients. Its resolution into factors is ren- 

 dered possible by the circumstance that only two of the quanti- 

 ties <£, ^r, d . . . can bear a finite ratio to each other for any given 

 value of x, and consequently all the roots of the equation 



(<^)**+ (^xy+ [&*)'... =o* 



are contained among the roots of the several binary equations 



((f>x) i =(ylrx) i f (cj)%y={$a;y, &c. : 

 which are the roots of any one of these equations (as ex, gr. of 

 the first) that belong to the given equation will be determined by 

 the condition that they must make the norms of all the other 

 functions {ex. gr. of §x) indefinitely small as compared with the 

 norms of those two which appear in it (ex.gr. <j>x, ifrx). In this 

 manner, if the total number of the functions is k, supposing 



* My friend M. Jordan, of the Ecole des Mines (author of a remarkable 

 thesis on groups), has developed some interesting geometrical consequences 

 arising out of the study of this equation, which I hope he may be induced 

 to publish. 



