MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 73 



?4, m^, n^. Eliminating these from (59) there results the equation of 

 condition between a, j8, y, 8 that P may be at D, 



«= + i8= + 7=-8= 

 + 2apcosADB + 2PycosBDC + 2ayC0sCDA = 0. 



This equation asserts that there must be at Z* a direction acting along 

 which 8 is the resultant of a, 13, y along DA, DB, DC. Putting A, n, v 

 for the cosines of the face angles ADB, BDC, CD A, the locus of points Q 

 for which P is at i? is the hyperboloid of two sheets 



x--\ry- + Z' — w- + 2xyX + 2yztt. + 2zxv = 0. (69) 



This surface passes through the mid-points H, I, J, of f, g, h. It does not 

 meet the plane w = 0, since the resultant of any numbers x, y, z along 

 f, g, h can not vanish. It cuts the faces A, B, C respectively in the hyper- 

 bolas which have already been constructed for the plane. Corresponding 

 to each vertex there is a similar hyperboloid passing through the mid- 

 points of the edges concurrent there. These hyperboloids serve to map out 

 the regions within which the position of Q gives s an ordinary maximum 

 or minimum value. For a position of Q on the boundary of this region 

 the equation to the surface in the form (68) will show, as was done for 

 the plane, that for Q inside the tetrahedron and on (69) the function s 

 has a minimum. 



We go no further into the detailed criticism of the singularities than to 

 observe that the investigations show that the tangent planes at the corners 

 of the tetrahedron to the circumconicoid 



S = xyc + xzh -\- yza -j- xwf ~\- yivg -\- zivh, 

 are respectively 



as ^ dS ^ dS ^ dS _^ 

 8x dy dz dw 



These planes form a tetrahedron, and it appears that when ^ is at a 

 corner of this, then the locus of the critical point P is the spherical cubic 

 whose equations are 



"Sixyzl- = 0, %xyc^ = 0, 



and when Q is at the vertex opposite to that of D, the radial equation of 

 that corresponding part of the spherical cubic on the sphere is 



ar, + pr„ -f yr. -f Sr^ = 0, 



