newson: unicursal curves by method of inversion. 67 



and the polar lines of the origin, by Uj-f-4UQ=o. 



The origin may be any point in the plane and hence we conclude 

 that only one quartic of the system passes through a given point and 

 that the polar cubics of any point form a system through nine points. 

 The polar conies of any point form a system through four points and 

 the polar lines meet in a point. 



If one of the critic centres be taken for origin, we can readily see 

 that such a point is also a critic centre on each of its systems of polar 

 curves. It is thus at a vertex of the self-polar triangle of its system of 

 polar conies and the opposite side of the triangle is the common polar 

 line of the critic centre with respect to each of the systems of curves. 

 The tangents at the node of the nodal quartic coincide with those of 

 its polar cubic and these we know coincide with the lines which con- 

 stitute its polar conic. 



If two of the sixteen basal points coincide, such a point is a critic 

 centre. The argument is the same as for a system of cubics. We can 

 also see that two of the basal points of each of its systems of polar 

 curves coincide at the critic centre. The sixteen basal points of the 

 system of quartics may unite two and two so that it is possible to draw 

 a system of quartics touching eight given lines each at a fixed point. 



,If three of the basal points of our system of quartics coincide, all 

 the quartics have at such a point a common point of inflection and a 

 common inflectional tangent. The demonstration is the same as that 

 already given for cubics. The system of polar cubics of such a point 

 also have this point for a common point if inflection and the same 

 tangent for a common inflectional tangent. I prefer to show this 

 analytically for the sake of the method. The equation of the system 

 of quartics having the origin for a common point of inflection and the 

 axis of y for a common inflectional tangent may be written 



u,+U3+^ (B+kB0xy+(C+kCJy2 ^+(A+kAJy=o. 

 The equation of the polar cubics of the origin is therefore, 



U3 + 2^ (B+kBJxy+(C+kCOy^ [^+3(A+kAJy=o, 

 which proves the proposition. The properties of the system of polar 

 conies of such a point are therefore the same as those already proved 

 for cubics. One quartic of the system has a double point at the 

 common point of inflection of the others. 



When four basal points coincide they give rise either to a common 

 point of undulation or a common double point on all the quartics of 

 the system. The equation of the system having a common point of 

 undulation may be written 



