1871.] Br. J. Casey on Cyclides and Bjjhero-Qucaiics. 



497 



into the evolute and the focal conies of a bicircular quartic. The deve- 

 lopable possesses numerous anharmonic properties ; thus all its gene- 

 rators are divided homographically by the nodal lines and the sphere U. 



In the chapters on the inversion and classification of cy elides, I have 

 proved that the presence or absence of nodes depends on the relative posi- 

 tions of the focal quadric and sphere of inversion ; thus if they touch 

 there will be a conic node, the cyclide being in this case the inverse of a 

 quadric, which is an hyperboloid or ellipsoid according as the node has a 

 real or imaginary cone of contact. If they osculate, the cyclide will be the 

 inverse of a paraboloid ; the node will be biplanar if the paraboloid be an 

 elliptic or hyperbolic one, and it will be uniplanar if the paraboloid be 

 cylindrical. If the focal quadric and sphere of inversion have double con- 

 tact, the cyclide will be the inverse of a cone of the second degree, and will 

 have two nodes, which must be conic nodes. When a cyclide has nodes, 

 the number of focal quadrics suffers diminution. I have given in the same 

 chapters the equations and the singularities of the tangent cones, and 

 shown that in general every cyclide has as many double tangent cones as 

 it has focal quadrics ; in fact the double tangent cones are the reciprocals 

 of the asymptotic cones of the focal quadrics. It is also proved that the 

 lines of intersection of a cyclide, with its spheres of inversion, are lines of 

 curvature on the cyclide, and that the imaginary circle at infinity is a 

 flecnodal curve on its surface of centres. 



In the chapter on the classification of sphero-quartics I have given 

 Chasles's characteristics for the osculating circles of a sphero-quartic. By 

 inversion we get the characteristics for the osculating circles of bicircular 

 quartics. Thus V= 24 for these circles. In the same chapter Professor 

 Cay ley's equations, giving the singularities of the cuspidal edges of deve- 

 lopables, are transformed so as to give the singularities of the evolute of a 

 plane curve, any three of the singularities of the curve being given. 



The last two chapters contain an account of the substitutions by which, 

 from properties of quadrics, may be inferred corresponding properties of 

 Gyclides. These chapters are in reality an exposition of a new method of 

 geometrical transformation ; in fact, since the general equation in a, j3, y, 2 

 which I employ is the same in form as the general equation of a quadric, 

 only that in my method the variables denote spheres in place of planes, it 

 will be readily seen that the theories of invariants, reciprocation, &c. in 

 the geometry of surfaces of the second degree have their analogues in the 

 theory of cyclides, and, in fact, the modes of proof employed in one apply 

 also in the other. This method of transformation is very fertile ; I have 

 illustrated it by numerous theorems. Thus the locus of the centre of a 

 variable sphere cutting in two sphero-quartics having double contact, two 

 cyclides having a common sphere of inversion is the developable circum- 

 scribed about the focal quadrics of these cyclides, which correspond to the 

 common sphere of inversion. 



VOL. XIX. 



