204 Mr. J. J. Sylvester on Poncelet's approximate 



cussion of the form v^a 2 — £ 2 ), attendu que dans les appli cations 

 de la mecanique aux machines les radicaux de la forme \Zfl 2 — 6 2 

 sont rarement a. considerer. Nous en dirons autant de ceux de la 

 forme VW^W+c^, qui represented la resultante de trois forces 

 rectangulaires entre elles et situees dans Pespace. D'ailleurs, 

 si l'on connait les limites entre lesquelles demeurent compris les 

 rapports des composantes a, b t c, ou de leurs resultantes par- 

 tielles >/fi 2 + & 2 , &c, on pourra toujours ramener ce cas au pre- 

 mier de ceux que nous avons examines," meaning to the case of 

 Vdt + b*. Now, in the first place, it is not clear how this 

 reduction can be effected in general, or indeed in the vast ma- 

 jority of cases that might be proposed. For instance, if we have 

 given a < Vb* + c 2 , a > b> a > c, I do not see how after, accord- 

 ing to M. Poncelet' s process, V a? + b q -\- c 2 is put un der the 

 form aa-f/3 Vb 2 -\- c 2 by aid of the limit a < \^b 2 + c 2 , any use 

 can be made of the other limits a > b, a > c in further reducing 

 this to the ultimate form aa + a! fib + fi'fic. Or if we take the 

 still simpler case, where a, b, c are left unlimited, in whatever 

 way we attempt to proceed we shall obtain different approxima- 

 tions, according to the order in which we effect the successive 

 reductions. 



Furthermore, in those few exceptional cases where the process 

 indicated by M. Poncelet leads to the use of all the limits given, 

 the form arrived at is not and never can be the true best form, de- 

 fined as such, according to M. Poncelet's own principles, as that 

 which within the given limits has its maximum proportional error 

 the least possible. Thus M. Poncelet indicates as the linear form 

 for Va 2 + b* -f c 2 , when the given limits are a 2 > b 2 -f c 2 , 6 2 > c 2 , 

 •96046a + '3820U + -15827c, with a maximum error textually 

 quoted from his memoir, *0507. It will be seen hereafter that the 

 true best linear form gives a maximum error about one-tenth less 

 than this. But it would be quite easy to give examples in which 

 the maximum error by Poncelet's process should exceed in an 

 indefinite proportion the necessary maximum error. This, for 

 instance, would be the case if we imposed the limitations 



x^ + y^Xz*, y* + z*>\a:*, * 2 + tf 2 >\y 9 , 



on taking X inferior but indefinitely near to 2. 



The geometrical method of demonstration given by M. Pon- 

 celet for the case of two variables, labours under the incon- 

 venience of beginning with a figure of three dimensions, and 

 consequently does not admit of being carried beyond that case, 

 although the result for three variables geometrically stated, when 

 the conditions of the question are set under an appropriate form, 

 are precisely analogous to that obtained by M. Poncelet for two 



