linear Valuation of Surd Forms. 217 



amples, that may be left to those who feel the want of the 

 Tables which this method affords. If the limiting conditions 

 were supposed to be -z > y, z> x } this would correspond to the 

 quadrilateral Z K' Q K in the last figure : it may easily be ascer- 

 tained that a circle passing through K'ZK would contain Q, 

 and would have its centre between N and Z. Hence by the 

 application of Peirce's law, we know that the minimum circle in 

 this case is that which can be drawn through K' Z K, and con- 

 sequently the linear form and maximum error will be precisely 

 the same as for the simpler case already considered, z >\/ 'x^ + y*. 

 On the other hand, if the conditions imposed were simply z < x, 

 z <y (conditions, be it remembered, far wider than ever would be 

 admitted in practice), the limiting figure becomes XQY; and 

 since MQ < MX or MY, the centre of the circle through XQY 

 would fall under X Y, so that the limiting circle in this case 

 would be that having M for its pole; the linear substitutive 

 form would not contain z, but would be the same as if z did not 

 appear, viz. -96046# ■+ *960467y, with -03954 as the maximum 

 proportional error. The same remark would apply to the system 

 of conditions z < \x, z<\y for any val ue of X not inferior to ^/\. 

 The conditions z> x, z>y, z < \J ' x 1 + y 2 would correspond 

 to the limiting area K K 7 Q, which would give rise to the deter- 

 mining matrix, 



The condition z < s/# 2 + y 1 would correspond to a limiting area, 

 KK'XY. If KY be bisected in G, and K'X in G*, and G'YGX 

 intersect in H, it is obvious that a small circle may be de- 

 scribed with H as its pole passing through all four points 

 X, Y, K, K ; , which will be the minimum circle of limitation. 

 To assign the determining matrix, we may take any three of 

 these four points, as, for example, Y, X, K, which will give 



This gives 



Q=V5 



F =v /J, G=V% H = l-Vf= -29289, 

 N 2 =4-v / 2=l-085786, 

 N = 104200, 



N + Q=l-74911, N-Q=-33489. 

 Phil Mag. S. 4. Vol. 20. No. 132. Sept. 1860. Q 



