45 



F and J, then, if a variable S, cutting J orthogonally, has its centre on 

 F, its envelope is a Cartesian oval. The centre of F will be the triple 

 focus of the oval ; and the three single collinear foci will be the centre 

 of J, and the two limiting points of J and F. I have shown, * also, 

 that the oval has six other foci, which lie two by two on three lines 

 perpendicular to the line of collinearity of the single foci. 



I have entered at some length into the properties of circular cubics. 

 All the properties of these curves which I give in this paper I believe 

 to be new. Thus, "being given four con cyclic points," I have proved that 

 " the two circular cubics which can be described having these points as 

 single foci are such that the point where each intersects its asymptote 

 is the double focus of the other ;" and, again, that " the circle which has 

 the distance between these double foci as diameter is the ' nine points' 

 circle' of the triangle formed by any three of the four centres of inver- 

 sion of either." 



I have next discussed the characteristics of the various curves treated 

 of in the paper, and of their evolutes, not only determining them for 

 the quartics and cubics of each class, but showing the exact points and 

 lines which are cusps, double tangents, stationary tangents, &c. and 

 have arrived at some new theorems respecting the osculating circles of 

 conies as well as bicircular quartics. Thus, " through any point not 

 on an ellipse or hyperbola can be described six circles to osculate the 

 ellipse or hyperbola, and through any point not on a bicircular quartic 

 of the eighth class can be described twelve circles to osculate the 

 quartic." 



Avery considerable portion of the paper is occupied with the appli- 

 cation of the methods of conies to bicircular quartics. In fact, since 

 the general equation of the second degree in a, ft, y which I employ 

 is the same as the general equation of a conic, only that in my method 

 the variables denote circles in place of lines, it will at once occur to 

 any one that the methods used in the higher parts of conies apply also to 

 bicircular quartics. I have entered very fully into this part of the 

 subject, and have shown that the theories of invariants and covariants, 

 reciprocation, and anharmonic ratio in conic sections, not only have 

 their analogues in bicircular quartics, but that the very same equations 

 and modes of proof which are employed in the one hold also in the 

 other. In fact, this part of the paper may be regarded as an exposition 

 of a new method of geometrical transformation ; and it is shown that 

 every graphic property of a conic section has an analogous property 

 in bicircular quartics. Thus, "The four conies having double contact 

 with a given conic U, which can be drawn through three given points, 

 are all touched by four other conies having also double contact with £7." 



Corresponding to this we have the following theorem in bicircular 

 quartics : — " The four bicircular quartics having quartic contact with a 

 given bicircular quartic which can be described so as to have double 

 contact with three given circles, have all double contact with four other 

 bicircular quartics having also quartic contact with U" 



I intend to follow up the mode of investigation employed in this 

 paper in kindred parts of geometry. 



