428 



Problem. To analyse the forms assumed by the locus of the 

 general eq. 



then 



Result. LetV= 



A D E 

 DBF 

 E F C 



andW= 



^+2Fy.iy+G=0 (axes oblique). 

 A D E A 2 

 D B F B 2 

 E F C C 2 

 A 2 B 2 C 2 G 



the common treatises (only without this notation) it is shown that 

 when V is finite, the surface (if real) has a centre. It is here added, 

 that when W is negative, the curvature is everywhere towards the 

 same side of the tangent plane ; when W vanishes, the tangent plane 

 coincides with the surface in one straight line ; but when W is posi- 

 tive, the surface is cut by the tangent plane in two intersecting 

 straight lines, and the curvature bends partly towards one side of the 

 tangent plane, partly towards the other. 



Hence it appears that we have different sorts of surfaces, by com- 

 bining V=0 or V= finite, with W=0 or W= positive, or W= nega- 

 tive. 



The locus is imaginary, if "W is >0, A and B finite, CG C 2 2 >0, 



A E A 2 



andC 



E C C 



A 2 C 2 G 



The locus is degenerate, if of ABC one at least (as C) be finite, 



j A E A 2 | B F B 2 



andifV=0, | E C C 2 =0, F C C 2 = : or if AB C all 

 i A 2 C 2 G I B 2 C 2 G 



vanish, and if at the same time D=0, and E : F : C 2 =2A 2 : 2B 2 : G. 



Problem. To investigate the nature of the plane intersections of 

 the surface. 



Result. If the cutting plane be lx + my + nz+p=^Q, the section is 



A D E I 



a hyperbola, parabola or ellipse, according as 



tive, zero, or negative. 



D B F m | 

 E F C n 

 I m n o 



is posi- 



