248 PROCEEDINGS OF THE AMERICAN ACADEMY 



But this can be reduced to the form 



S2;i^ _ SI^SIq — SUSXq + BX^SXd = 0, 



for we only assume 



/* = SX^ + Sid 

 and 



c = — 8l§S).d, 



and these are but two conditions to determine two unknown quantities. 

 Our equation now becomes 



(SXq — SX§) (SAo — SXd) = 0, 



and may be decomposed into 



SX(n — §) = 

 and 



s;.(c) — 5) = 0. 



Two parallel planes perpendicular to X. 



CiRCULAU Sections. 



Almost the only things worthy of notice about the paraboloids are 

 their planes of circuhir sections ; and these are interesting chiefly on 

 account of their connection with the planes of circular sections of the 

 central quadrics. 



Differentiating the equation of the paraboloid 



we obtain that of its tangent plane 



2 c.^Sa./jSu./y -\- 2 CgSftgoSajg' = 2kSaiQ'. 



At the origin, q vanishes, and dropping the accent of q', we get for 

 the tangent plane 



S«jP = 0. 



Now if we find a sphere with the same tangent plane, obtain the 

 equation of the surface of the second order in which lies the intersec- 

 tion of this sphere with the paraboloid, and finally discover the con- 

 dition that this surftice should degenerate into a pair of planes, we 

 shall have the planes whose intersection with the sphere, and conse- 

 quently with the paraboloid, will be circles. 



