Idea of Poncelet's Theorem, 315 



which will easily be seen to be a cubic (not a biquadratic) equa- 

 tion in x. Call (^y — 1 ) =<£(#) ; then the three roots of this 



equation being named x Xi x 2f x 3 , the law of equality explained in 

 my preceding paper would seem to show* that we must be able 

 to satisfy the following equations, 



(<K) 2 = (<K) 2 = (<K) 2 = (*«)*= (**)'. 



which amount to four independent equations, the precise num- 

 ber of constants X, p,, v } q to be determined. So in like manner 

 the ith rational approximant will contain 2i disposable constants ; 



the differentiation of the quantity analogous to ^r will give rise to 



an equation of the (2i — l)th degree ; and there will be 2i — 1 -f-2, 

 that is, 2i + 1 functions of these 2* quantities to be equated, 

 which furnish precisely the required number of equations to 

 make the problem definite. It is, however, apparent that in 

 solving these equations we shall find a multiplicity of systems, by 

 which I mean a definite number of systems of values of the dis- 

 posable constants which will equally well satisfy the equations. 

 For instance, in the theory of the second approximation, the 

 equalities (^ 1 ) 2 =(^ 2 ) 2== (^3) 2 w ^ De satisfied by supposing 

 x l -=Xc L -=x 3 \ . But it is by no means evident a priori that this 

 system of equalities will correspond to the absolute minimum of 

 which we are in quest : nay, though even we had (px 1 = </># 2 = <f>% 3) 

 those equations do not necessarily imply x l = x 2 =x 3 . Of the 

 multiplicity of solutions referred to, one only gives the true 

 minimum ; but to assign a priori the distinguishing marks of 



* Is it not, however, somewhat uncertain whether the equalities 



must all, in all cases (that is to say, for all given values of the limits) sub- 

 sist ? since the law of equality will not apply to such values of x as lie 

 without the prescribed limits, and non constat a priori that the roots of the 

 cubic do all lie within these limits. The subject at the very threshold is 

 beset with doubts and difficulties of a peculiar kind, which we can hardly 

 hope to overcome without calling in geometrical imagination to our aid. 



T If this is so, we shall have for determining the four constants the fol- 

 lowing equations : 



x 1 =x 2 -=x 3 , (pa=(pb= — (/)#!• 



But more probable than this seems the conjecture, that, supposing X\, %»> x * 

 to be arranged in the order of their relative magnitudes, the determining 

 equations might be 



xi=x 3 , (£a=<£&=(/># 2 = — (f)X v 



Or is it possible that the character of the solution may be discontinuous, 

 and may depend upon the magnitudes, relative or absolute, of the given 

 limits a and b ? Probably Dr. TchebitcheiF would be able better than 

 any other living analyst to answer these queries. But what an endless 

 vista of future research does the prosecution of the Ponceletic method 

 open out to us! 



Y2 



