178 Dr. Booth on a Theorem in Analytic Geometry. 



Now the triangles PCD, P Q r are similar, hence 



PD:PC::Pr:PQ, orZ:C::y:8, hence 



z y • im x u y Q 



— = 4-; in like manner — as -5-, 4- = •£-; 



c(> a 



making these substitutions in (l.)» we find 



a? y* z* „y z , x x z „xy i'tJ.\ 



T + 72 + -5 + 2 / cos A + 2 — Cos p + 2 -i COS V = 1 (2.) 

 or o* c z be ac ab v ' 



The equation of an ellipsoid whose centre is at the origin when 

 the coordinate planes are rectangular, the equation becomes 

 simply 



«2 T /)*» + C 2 - *' 



It follows immediately from (2.) that the coordinate planes 

 can never be conjugate planes of the surface, except when rect- 

 angular, as in no other case do the rectangles vanish. 



To find the coordinates of the point where the tangent plane 

 is parallel to one of the coordinate planes, that of xy. Sup- 

 pose V = 0, being the equation of the surface, the general 

 equation of the tangent plane is 



dV , ^ d\ . n dV , '"'""■ 



^(*-^) + ^Q/-y)+^(z-*') = 0; 



and when the tangent plane is parallel to that of xy, 

 Now 



— =• — = 



dx ~ * dy 



dV x y z 



-j— — f- ■— cos v -\ cos a = 0, 



dx a c 



dV y x z 



-r— = 4- -1 cos v -I cos a — 0: 



dy b a c 



from these equations, finding the values of x and y in terms 

 of z, and substituting in (2.), there results 



2 /" 1 — cos 2 A — COS 2 jU. — cos 2 v -f 2 cos X cos JO. cos v~\ _ 2# 

 L sin 2 v J 



the expression within the brackets is the square of the sine of 

 the angle which the axis of z makes with the plane of xy, 



calling this angle <p, z = - — ; now this value of z is evidently 



a conjugate diameter to the plane of x y, since the tangent 

 plane is parallel to the plane of x y ; hence whenever the ge- 



