222 On Poncelet's approximate Valuation of Surd Forms. 



spoiiding case in piano, with exhibiting a single numerical ap- 

 plication of the method. 



Suppose the given limits to be defined by the equations 



•* > | {y* +3?) y~>- #• 



Here it is obvious that the enveloping condition will be expressible 

 by means of the equation to a plane drawn through three points 

 on the hyperboloid, the coordinates of one of which are found 



And if we call the minors obtained by leaving out the first, 

 second, third, fourth columns respectively H, G, F, Q, the linear 

 form becomes 



2Hz 2Gy 2F# 



V / H 2 -G 2 -F 2 + Q + v / H 2 -G 2 -F 2 + Q + v /H 2 -G 2 -F 2 + Q 



s£ . Q_ v /H 2 -G 2 -F 2 A . 



with a maximum error ~ . =. And since 



Q + v / H 2 -G 2 -F 2 



Q=-v/2, H =v /2, -Grrx/3-1, -F= (\/2 -])(v/3-l 



we have 



V /H 2 -G 2 —F 2 = 11714 and Q= 1*4142, 



so that the representative form becomes l # 093ar — *566y— -089z, 

 with a maximum relative error of about -094. 



