228] FREEDOM OF THE FOURTH ORDER. 245 



cylindroid reciprocal to the enclosing four-system. It has also eight singular 

 tangent planes (four through each double line) touching all along a generator. 

 The cylindroid is a very special case of this surface in which one of the 

 double lines is at infinity. If the system be reciprocal to the cylindroid 

 z(x 2 + y f ) + kx;y = Q, then the cylindroid to which the surface reduces is 

 z(as * + y 2 ) kxy = Q. The singular tangent planes are represented by the 

 two tangent planes at the limits of the cylindroid. 



