linear Valuation of Surd Forms. 215 



or say 



•73025a? + '73025?/ + '73025*, 



with a maximum proportional error 



^t-\ or 2-^/3 = '26895. 

 v 3 + 1 



The corresponding error for \/x 2 + y 2 under the form -8284# 

 + •8284?/ is -17160, or about two-thirds of the one in question*. 



Ex. 2. z >- \/ 1/*- + x 1 . Here the determining matrix is 



F = G=v / |-i = -207107 

 H=4 



Q=i 



N 2 =F 2 + G 2 + H 2 =l- v / I=-292893 



N = -541196 



N + Q=1041196 N-Q=041]96. 



Thus the linear approximation becomes 



•397825a* + -397825?/+ -960430*, 

 with a maximum error -039493. 



Ex. 3. z> \A/ 2 + # 2 , y>x. This is M. Poncelet's example 

 (Crelle, vol. xiii. p. 291) . His a, b, c correspond respectively with 

 my z, y, x ; there are some misprints in line 6 of this page (in 

 M. Poncelet's Memoir) which may perplex the reader; it is in- 

 tended to stand thus : 



8Va* + b* + c* + /3ZV¥Tc*=Va* + b* + c*. fo+^x/ ^ + y + gJ "' 



Here the determining matrix corresponds to the areaZ KN (the 

 coordinates of N being found from the equations * 2 =a? 2 -f y 2 , 

 y=x, ,2' 2 + t r 2 + 2/ 2 =l), and the matrix will be as subjoined. 



* It would have been more exact to have treate/1 this as a case of a circle 

 to be drawn through four points, viz. Z the middle points of ZX, ZY and 

 the middle or lowest point (in reference to Z) of the small circle drawn 

 through these two, and having Z for its pole. But it is easily seen that 

 the small circle drawn through the three former will contain the one last 



— 45 



named, for the tangent of its circular radius will be V 2 X tan-^ , and con- 

 sequently its summit will be further from Z than from the point in question. 

 A similar remark applies to the subsequent and some other examples. 



