710 PEOF. W. K. CLIFFOED ON ME. SPOTTISWOODE’S CONTACT PEOBLEMS. 
Locus of poles of axis in regard to four-point conics. 
If we select any plane through the line AC, there will be a singly infinite number of 
conics in the plane having four-point contact with the surface at A. The line AC will, 
as is well known, have the same pole in regard to all these conics — that is to say, the 
point B will be the same for the whole system. If we now allow the plane to turn 
round the axis, the point B will trace out a curve in the tangent plane. The equation 
to this curve is got by eliminating X between equations (3') and (4'); namely, it is 
0 = (B 2 )(BC) — (B 3 )(C). (4") 
We see, therefore, that the locus of the poles of an axis in regard to all the four-point 
conics whose planes pass through it is a cubic curve in the tangent plane touching the 
chief tangents at A, which point is therefore a node on the curve. 
We might have inferred this from the fact that on any line through A there is only 
one point B, while this point coincides with A in the case of the two chief tangents ; 
since at a point of inflexion all the four-point conics contain the inflexional tangent. 
Number of six-point conics through an axis. 
We have now to determine B so that the equations (3'), (4'), (5'), (6') may be simul- 
taneously satisfied. We know already that B must lie on the cubic (4") ; it is necessary 
therefore to find some other locus on which it has to lie. First of all, then, we must 
eliminate X between (3') and (5') and between (3') and (6') ; the results are, 
(B 2 )(C) 2 = (B 4 )(C) 2 — (B 2 C)(B 2 )(C) + (B 2 ) 2 (C 2 ) say ; 
(B 2 )(C) 2 =(B 5 )(C) 2 -(B 3 C)(B 2 )(C)+(B 2 ) 2 (BC 2 )=^ 5 , say. 
Here ^4=0 and ^ 5 =0 are curves touching the chief tangents at A, and of the degrees 
four and five respectively. But the equations are not homogeneous ; in fact only the 
ratios and not the absolute values of the quantities a were determined by the fixing of 
the point A, and they may be regarded as involving an arbitrary factor whose square 
aflects the left-hand side of the equations. It is, however, at once eliminated, and we 
obtain the homogeneous result, 
(B 2 ) . S' 5 =(B 3 ) .p t . 
This is a curve of order 7 having two branches in each of the chief directions at A. Of 
its 21 intersections with the cubic (4"), then, 12 coincide with A, and there remain nine 
positions of B which give sextactic conics through the fixed axis ; or we may say, of the 
sextactic conics at the point A, there are nine whose planes pass through an arbitrary 
point C. 
System of five-point conics through an axis. 
Since there is one five-point conic in every plane, if we consider all the planes through 
a fixed axis we shall obtain a singly infinite number of five-point conics. Of this 
