linear Valuation of Surd Forms. 211 



with a maximum proportional error 



N-Q 



N representing 



The second case is where the limiting plane has to be drawn 

 through two points upon the sphere so as to cut it in a circle, 

 of which the line joining the two points is a diameter. 



In this case, calling the coordinates of the two points respect- 

 ively a, /3, 7; a', 7', and writing «a' + fi/3' + 77' = m, it is 

 easily seen that the perpendicular upon the limiting plane is 



— ^r— , and consequently the perpendicular upon the plane 

 20 



L* + My + N*=l is iii + v /l + w \. 



Also this plane being parallel to the limiting plane, is perpendi- 

 cular to the line joining the origin to the point 



<"!"*;" 2 " 2 ' 2 J 



and therefore 



j^kM, M =£±£, M*Ma5, 



P P P 



and 



I - -'-Ti gi-/5gV< 



v /(« +a ') 4 +(/3+/3') 2 + ( 7 + 7 ') 2_J I + V 2 J' . 

 that is to say, 



= H^(l+m)4-(H-m)}i 

 so that the linear form required is 



{y2(l+m) + l + m}{_^ * + fi^fr^S^iJpj 

 with a maximum proportional error 

 \/2—\/l+m 



\/2 + x/l+m 



(m is of course identical with the cosine of the angle between 

 the radii joining the two given points.) 



The conditions of inequality which obtain between x, y, z may 

 be, and usually will be, such as correspond to the limitation of 

 the point (#, y } z) to an area contained within a triangle or 



