230 PROCEEDINGS OF THE AMERICAN ACADEMY 



if all the c's are negative, there are no real semiaxes, and we get the 

 so-called imaginary ellipsoid. 



It is well known (Ham. Elem., § 354; Tait, §§ 163, 164) that the 

 three c's are the roots of an algebraic cubic, and are always real. We 

 shall find it convenient to take these roots in the algebraic order: 

 Cj<C2<C3. The general scalar equation, 



SpqpCJ = 1, 



where g) is self-conjugate, may be written in the rectangular form : — 



c^^Hq + C2S7P + CgSU-^ = 1. 



K any two roots, as c^ and C3, are equal, a plane 



Sip = m 



perpendicular to the third direction {i) cuts the surface in the circular 



section, 



CgS^'o -j- c.^^-hn =1 — qm^. 



So the quadric is a surface of revolution. If all the roots are equal 

 — and this, of course, can only happen in the case of an ellipsoid, — 



c^^Hq -\- c^S-Jn -|- CjSU'o = 1, 



and therefore SHq 4- S^'p + S^kg = —, 



'-1 



or, T^Q (cos2< '■ + cos2< ^' -{- cos2< ^■) = T^q = I, 

 and Tp = -p . 



The surface is then a sphere with c^ — i for a radius. 



If the constant term vanishes, we have already seen that the surface 

 must be a cone. Neither of the principal diameters can be in the 

 direction of a side of the cone ; for, if 



Q =z zi 

 for instance, we find 



c^SHxi = x^Cy = ; 



and therefore x = 0. 



