NEWSON: UNICURSAL CURVES BY METHOD OF INVERSION. 53 



respect to z; we obtain thus the equations of the polar conies and polar 

 lines of the origin with respect to the system. 



The polar conies of the origin are given by 



3(a+kaO+2J (b + kbjx + (c+kci)y [ +U3=o: 

 thus showing that the polar conies of any point, with respect to the 

 system of cubics, form a system through four points. The polar lines 

 of the origin are given by 



3(a+kaO +(b+kbOx + (c+kcOy=o, 

 which represents a pencil of lines through a point. 



Suppose now the origin to be at one of the critic centres; then for 



a particular value, kj, all terms lower than the second degree must 



lla b c II 

 vanish, so that , r^o- The factors of the terms of u„, 



11^1 b^ cjl 



which involves kj, represent the tangents at the double point to the 

 nodal cubic, and also the polar conic of the origin with respect to this 

 nodal cubic. Hence a critic centre is at one of the vertices of the 

 self-polar triangle of its system of polar conies. The opposite side 

 of this triangle is the common polar line of the critic centre with 

 respect to its system of polar conies, and hence it is also the common 

 polar line of the critic centre with respect to the system of cubics. 

 The four basal points of the system of polar conies lie two and two 

 upon the tangents at the double point of the nodal cubic. 



When the origin is taken at one of the nine basal points of the system 

 of cubics, a and a^ both vanish. Hence it is readily seen that a 

 basal point of a system of cubics is also a basal point of its system of 

 polar conies and the vertex of its pencil of polar lines. 



Suppose two of the basal points of the system of cubics to coincide, 

 then every cubic of the system, in order to pass through two coincident 

 points, must touch a common tangent at a fixed point. The common 

 tangent is the common polar of its point of contact, both with respect 

 to the system of cubics and to its system of polar conies. Hence the 

 union of two basal points gives rise to a critic centre. The self-polar 

 triangle of its system of polar conies here reduces to a limited portion 

 of the common tangent. This line is not a tangent to the nodal cubic, 

 but only passes through its double point. 



Suppose three of the basal points of a system of cubics to coincide, 

 such a point will then be a point of inflection on each cubic of the 

 system. For, in the last case, if a line be drawn from the point of 

 contact of the common tangent to a third basal point of the system, 

 such a line will be a common chord of the system of cubics. Suppose, 

 now, this third basal point be moved along the curves until it coincides 

 with the other two; then the common chord becomes a common tang- 



