392 Royal Society :-— 
which are the test, that the plane may touch a sphere given in 
position. 
Problem. To analyse the forms assumed by the locus of the 
general eq. . 
2 Aa’ + By? + C2 + 2A,e+ 2B,y + 2C,¢ 
+ 2Dey+2Exz+ 2Fyz+ G=0 (axes oblique). 
ACD ADDER A, 
Resulé. Let V=|D B F/ and W= DBFB, ; then in 
EFC EFCC, 
A, B,C,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, CGQ—C,2>0, 
AEA, 
and C| E C C,| >9. 
A, C, G 
The locus is degenerate, if of ABC one at least (as C) be finite, 
AE A, BF B, 
and if Y=0,|E C C,|=0,|F C C,j =0: or if ABC all 
A, C, G B, C, G 
yanish, and if at the same time D=0, and E: F: C,=2A,: 2B,:G. 
Problem. To investigate the nature of the plane intersections of 
the surface. 
Result. If the cutting plane be /v+-my+nz+p=0, the section is 
ADE 
. : DBF m 
a hyperbola, parabola or ellipse, according a . . 
YI p a Ipse cording as + pO 4 ore 
lm no 
tive, zero, or negative. 
ADE A,J 
DBF Bm 
| EFC C,n |=0. 
| 42 B,C, G p 
| mn po 
The intersection degencrates, if 
