linear Valuation of Surd Forms, 209 



and the perpendicular upon the tangent plane is 



4. 



V 



i/A*+B 2 + C 2 V A 

 Consequently the mean between this and the perpendicular upon 

 the given plane is 



1 y/P + y/A . 



VA 2 + B 2 +C 2 2«/A 

 and therefore the equation to the plane required is 



so that 



v/P + v/A * v'P + x/A \ s/Y+<s/± ' 



La? + My + N> being the approximate representation of V4Hp$y>z)i 

 and the maximum error being evidently 



\ZP + \/a" 

 These results are perfectly general, and apply to a quadratic 

 radical of an integer homogeneous quadratic function of any 

 number of variables ; thus for \/$(#, y, z, t) the linear repre- 

 sentative form is 



2^/A.A , 2\/A.B 2\/A.C , 2\/A.I> . 



v/P + x/A x/P + x/A* v/P + v/A x/P-fVA 

 and the greatest proportional error is still 



v/p-Va. 



x/P+x/a' 



D signifying the discriminant, and P the polar reciprocal of 

 *(A, B, C, D). 



For the sphere, the perpendicular upon any tangent plane 

 being 1, the linear form ought to be that obtained from the 

 equation Ax + By + Cz = K, where 



K _ 1 / 1 \ 



>v/A 2 + B 2 4-C 2 "" 2 \ + VW+W^C^' 



first sight appear strange that P should be of the form of a contravariant 

 (in lieu of a covariant) ; but it must be remembered that the axes to which 

 the line or surface and its chord are referred are supposed to be orthogonal, 

 and for orthogonal substitutions, contravariants and covariants are indi- 

 stinguishable. 



