250 PROCEEDINGS OF THE AMERICAN ACADEMY 



Now tbis represents two lines perpendicular to X and /* 



X=:±^COt(i0 +K> 



or (vide Fig. 3, page 246) 



x = zcot(h + <^), 

 and 



X=ZCOtaQ)+ <l). 



Introducing j with an indeterminate co-efficient, we have two planes 

 perpendicular to X and jw, cutting the surface in circular sections. And 

 this result we should be led to expect from the fact that the paraboloid 

 is a limiting case between the ellipsoid and the hyperholoid. 



Tills one example is sufficient to show that, with the proj^erty of 

 self-conjugation, the general equation of the second degree loses that 

 simjjlicity of expression which makes quaternions so singularly a[)pli- 

 cable to the central quadrics. Tbe very form of the self-conjugate 

 function exliibits some of the fundamental properties of these central 

 surfaces. The rectangular transformation depends on the principal 

 axes ; the cyclic, on the relations of the cyclic normals ; while the 

 focal shows at once the properties of the focal lines of the asymptotic 

 cone. Other transformations could doubtless be discovered, embody- 

 ing other well-known properties of these remarkable surfaces. 



Cambridge, May 28, 1877. 



