Fraser — Reduction of a Quartic Surface to a Canonical Form. 7'i 



This denotes then a system of quadrics possessing a Jacobian 

 sui'face. 



That is, tlie system, of quadrics passing through the nodal conic and 

 touching the quartic surface twice possesses a Jacobian surface, and 

 there are five such systems. 



J 



■ = - D^w'^lDx'' + f + z^) - 2w [Lx ^ My -Y xVz] + \Lw'\ 

 J contains w^ as a factor ; and the remaining surface is the quadric 



1w \Lx ^ My ^ N%] 



Dw 











Dx- 2ivL 







JDw 







Dy - IwM 











Dw 



B% - 2wN 



J)x 



By 



D% 



-XDw 



+ %"■ - 



B 



+ Xw" = 0. 



And there are five, and only five, such qnadrics. 



Consider the point which has w for its polar plane with respect to 



aM + /?«; + y«^ + ^ = 0, 

 where 



TJ = Bwx - w^L 



V = Bwy - w'^M, 



W = Bw% - w'^N, 



T = B\_x^ ^y"" \%^ - \w'\ 



B[aw + 2«] = 0, Bll^w + 2y] = 0, B^aw + 2z] = 0, 



ic, y, z, w being the coordinates of this point ; hence if, in iii., we 



substitute for a, /?, y. 



respectively, we get 





x 



y 



% 





~w' 



w' 



'w' 



w^ 



X 



y 



% 



X 



a + 2X 



h 



9 



y 



h 



h + 2\ 



f 



= 0, 



z g f c + 2\ 



as the condition which the coordinates must fulfil — that is, the locus 

 of the point is a quadric. 



X + y'^ + z^ 



Ji = 



B, 



{Zix + M^y + JViz) + \ 



i«^' , 



