16 



RICE 



ART. B 



when the origin of the axes is at the centre of the surface. It 

 can be proved that the equation of the plane which is tangent to 

 the surface at the point Xi, y\, Zi on the surface is 



(axi + %i -\-gZi) X + (/ixi + hyi + fzi) y 



+ (gxi 4- fyi + czi) z = k. 



Hence the direction-cosines of the normal to the surface at the 

 point Xi, yi, Zi are proportional to the three expressions 



aXi + hyi + gzi, hxi + byi + fzi, gxi -\- fyi + czu (10) 



Another result which is required concerns the changes in the 

 coefficients in the equation of the surface if the axes of reference 

 are transformed to another set of three orthogonal lines meeting 

 at the centre. If the coordinates of a point are x, y, z referred 

 to the old axes and x', y', z' referred to the new, the values of x, 

 y, z can be worked out in terms of x', y', z' and the nine direc- 

 tion cosines of the new axes with reference to the old. On put- 

 ting these values for x, y, z in the above expression, we obtain the 

 equation of the quadric surface referred to the new axes as 



a'x'^ + by^ + c'z'^ + 2f'y'z' + 2 g'z'x' + 2 h'x'y' = k, 



where the values of a', h', c',f', g', h' can be obtained in terms of 

 a, h, c, f, g, h and the nine direction cosines. The following 

 three results can then be proved : 



a' -\- b' + c' = a -}- b + c, 



b'c' + cW + aV - P - g'^ - h"" 



= be + ca -\- ab — p — q^ — h^, 



\iM) 



The interested reader will find the proof in any standard text 

 of analytical geometry. 



