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



The linear approximation is accordingly 



•8090* + -8090^ + '335 Is, 



with a maximum proportional error '1914. 



Finally, for z > y, y>x the limiting triangle will be Z K Q, 

 the determining matrix 



1 T 



S\ Si 1 



•i ^y ^i 1 



F=v / |-v / i = '1297, 0=^(1- •*} ='1692, 



H=v / J=*4082, 



N 2 = f-VI-\/f=: '21207, 



N=4605, N + Q=-8687, 



Q= v /| = -4082, N-Q=0523. 



The linear approximation is •2986#+ , 3895y + " 9397y, with a 

 maximum error *06 (more precisely -0602). This is a trifle 

 beyond half as much again as the maximum error of the best 

 linear approximation to isjx^ + y 9 -, subject to the limitation x>y, 

 which (see Poncelet's Memoir, p. 280) is a little under '04. 



Poncelet has shown that for Sx 2 + y 2 } when x, y are the co- 

 ordinates of a point limited within a sector whose bounding 

 radii make angles <f> and yjr with the axis of X, the approximate 

 linear form is 



cos 2 . cos 2 , 



with a maximum error tan 2 - , . 



4 



In like manner it follows immediately from the method given 



in the text, that if the summit of the limiting segment make 



angles X, p, v with the axes of X, Y, Z, and its sp herical radius be 



p, the approximate expression for Vx^ + y^ + z 2 is 



cos X cos Lb cos v 

 x+ —y+ z, 



«P aP 9P 



COS* ^ COS 2 § COS 2 § 



A Z A 



with a maximum error tan 2 £, which expressions are the precise 

 analogues of the former, as will immediately appear from the 



