PROFESSOR CAYLEY ON POLYZOMAL CURVES. 55 



Centre of the Circular Cubic, and Nodo-Foci,&c. of the Bicircular Quartic — Art. Nos. 127 to 129. 



127. The tangents at J, J have not been recognised as tangents from I, J, 

 giving by their intersection a focus, but it is necessary in the theory to pay 

 attention to the tangents in question. It is clear that these tangents are in fact 

 asymptotes — viz., in the case of the circular cubic they are the two imaginary 

 asymptotes of the curve, and in the case of a bicircular quartic, the two pairs of 

 imaginary parallel asymptotes; but it is convenient to speak of them as the 

 tangents at I, J. 



128. In the case of a circular cubic, the tangents at I and J meet in a point 

 which I call the centre of the curve, viz., this is the intersection of the two 

 imaginary asymptotes. 



129. In the case of a bicircular quartic, the two tangents at / and the two 

 tangents at J meet in four points, which (although not recognising them as foci) 

 I call the nodo-foci ; these lie in pairs on two lines, diagonals of the quadrilateral 

 formed by the four tangents (the third diagonal is of course the line IJ), which 

 diagonals I call the " nodal axes;" and the point of intersection of the two nodal 

 axes is the "centre" of the curve. The nodo-foci are four points, two of them 

 real, the other two imaginary, viz., they are two pairs of anti -points, the lines 

 through the two pairs respectively being, of course, the nodal axes ; these are con- 

 sequently real lines bisecting each other at right angles in the centre (with the 

 relation 1 : i between the distances). The centre may also be defined as the inter- 

 section of the harmonic of IJ in regard to the tangents at /, and the harmonic of 

 this same line in regard to the tangents at J. Speaking of the tangents as 

 asymptotes, the nodo-foci are the angles of the rhombus formed by the two pairs 

 of parallel asymptotes ; the nodal axes are the diagonals of this rhombus, and the 

 centre is the point of intersection of the two diagonals ; as such it is also the 

 intersection of the two lines drawn parallel to and midway between the lines 

 forming each pair of parallel asymptotes. 



Circular Cubic and Bicircular Quartic ; the Axial or Symmetrical Case — Art. No. 130. 



130. In a circular cubic or bicircular quartic, the pencil of the tangents from 

 / and that of the tangents through J, considered as corresponding to each other 

 in some one of the four arrangements, may be such that the line I J considered 

 as belonging to the two pencils respectively shall correspond to itself, and when 

 this is so, the four foci, A , B, C, D, which are the intersections of the correspond- 

 ing tangents in question, will lie in a line (viz., the conic which exists in the 

 general case will break up into a line-pair consisting of the line I J and another 

 line). The line in question may be called the focal axis ; it will presently be 

 shown that in the case of the circular cubic it passes through the centre, and that 

 in the case of the bicircular quartic it not only passes through the centre, but 



