Dr. Booth on a Theorem in Analytical Geometry. 445 



the three fixed points ; on the contrary, when P is between 

 any two of the points, the corresponding pair of rectangles 

 become negative. 



To determine the volume of this surface, let U = , be the 

 equation of a sphere, whose radius is r, referred to the same 

 oblique axes of coordinates, having its centre at the origin, 

 and touching the ellipsoid at one of its vertices; then if a 

 tangent plane to the ellipsoid be drawn at this point, it will 

 also touch the sphere, and we shall consequently have, the 

 equation of the sphere being 



U z=:x 2 +y' 1 +z--\-2yzcos'A-\-2xzcos p+2 xy cos »— r 2 = (3.) 



dV dXJ dV dV dV dV 



dz " dz 9 



(4.) 



dx doc dy dy' 



as the coefficients of the variables in the equations of the co- 

 incident tangent planes are identical; hence 



x y z x + y cos v + z cos a ■ 



-3- + -^tCOSV H COSjU. = - a - 



a 2 ab ac ~ r 2 



y 2 x 



■To + T~ cos A H t- COS V 



b 2 be ab 



Z CO 7/ 



-a H cosa + -f- COS A = 



c z ac ^ be 



or putting t — — , w = ^-, there results 



y + z cos A + x cos v . . , - \ 



** 9 > ' \ 5 ') 



z 4- x cos ju, + y cos A 



a 



t t COSfX. COS |U< __ t + u COS V + COS jU. 



"I t ~ r 



a b 



a c 

 cos A 



u t cos v 



b 2 ab be 



u + t cos v 4- cos A 



1 tfCOSjX wcosA _ 1 + ICOSfl + mcosA 



h ~b~c~ + ~bc~~~ 



(6.) 



cf o c oc tr 



From these equations, eliminating t and w, we find the cubic 

 equation, putting 



1 — cos 2 A — cos 4 ft — cos 3 v + 2 cos A cos ft cos v = A 2 , 



r 6 ~r! [Vsin 2 X + 4 2 sin 2 fi + c 2 sin 2 v - (b c cos 2 X + a c cosV + a i cos 2 i>) 

 + 2(ab + ac-\- b c) cos Xcos /x cos vl 



-(-L.rjVsin 2 \ + a 2 c 2 sin 2 u4-a 2 J 2 sin 2 j/ - aftc(acos 2 X + b cos- p -\- c cos* v) 



4- 2 a i c (a + ft + c) cos X cos /t cos v~\ 



-a 3 6V = . . 



(7.) 



