242 PECMTEEDINGS OF THE AMERICAN ACADEMY 



order that a quadric should degenerate into its asymptotic cone, and 

 this is because if with this value of c we transfer the origin to the 



centre, 



a 9 ^ 



== — — «2 — — a3, 



'-2 ''8 



we obtain an equation without any absolute term. If in the equation 



7 = — «^«i — ^"2 — ^'«3 



g or h vanish, when c^ alone of the roots disappears, it would merely 

 indicate that the origin was already in a line with the centre as far as 

 that direction is concerned. 



Suppose two roots, as c^ and c^, vanish, and also the co-efficient of y 

 in the direction of one of them, say d. The solution for the centre 

 will then give 



1 7 1 1 , 



o ^=. a«j — — <jr«2 — — kciy 



C\ c-i H 



The centre is thus at an infinite distance in one direction, and at an 

 indeterminate distance in another. This can only be true of a para- 

 holic cylinder. Let us see what the general equation of the second 

 degree will give us in this case. 



c^S^ftj^ -\- c^S'a^n -j- c^S^a^Q -|- 2 Sj'p = c 

 becomes 



CjS^KaO — 2 gSa./^) — 2 kSa^o = c. 



And by cutting this by planes perpendicular to «j, «2' ^^^ «3, it can at 

 once be proved to be 2i parabolic cylinder. By substituting 



k c -I 



G = C — — «3 + . ^ «2' 



^9 

 the equation can be reduced to its simplest form 



CgS^KaP — 2 gSa^Q =: 0. 



If, as a farther condition, we had 



ff=0, 



this last transformation would not be possible ; but the equation could 

 be reduced to the form 



C3SV3O = c, 



or 



