linear Valuation of Surd Forms. 



219 



consideration that the summit of the spherical segment corre- 

 sponds with the centre of the circular arc. 



As an example of the use of these formulae, suppose the given 

 limits to be 



x < v^ + s 2 , y < \Z* 2 -f-# 2 , x < \/# 2 + y 2 . 



If we bisect the quadrants X Y, YZ, 

 Z X in L, M, N respectively, the vari- 

 able point will be limited to lie in 

 LMN, and the base of the correspond- 

 ing segment will be the circle passing 

 through LMN whose summit will be 

 at E, the point where the perpendicular 

 to XT at L and the arc bisecting 

 the angle X meet. 



Here then we have 



p = LE, \=/*=j/=XE, 

 tan p = cos 45 = s/\, 



cotX= V\, 



cosp 



= ^h 



2 



'•- = £{1+ */f}, cos\= s/$, 



Hence the linear approximation is 



= -6356744(# + y+z), 



with a maximum proportional error 5— \^24 = '10102. 

 More generally, if we assume the system of conditions 



\/# 2 + 2/ 2 > cz } vy 2 + z 2 >- ex, vs 2 + a? 2 > cy, 



c being any number intermediate between 1 and \/2, if in the 

 figure annexed, we take tan ZK=tanZK' = c, and join KK'by 

 a small circle intersecting YM which 

 bisects Z X in R, remaining still the 

 summit of X Z Y, it is easy to perceive 

 that the limiting area will be included 

 within the triangular space cut out be- 

 tween K K' and the two other analogous 

 small circles ; X, p, ¥ will remain the 

 same as before, and R will represent 

 p. Accordingly we have from the qua- 



Q2 



