Fraser — Heduction, of a Quartic Surface to a Canonical Form. 77 



which shows that the five quadrics belong to the same confocal 

 system. 



Z^ + M^ + W . A + B+ C-X.J) /'(A) 



^ = IP ^ = ^ = ^r ' 



and hence, in this case, the identical linear relation becomes 



T '^ 



xY + z^ [{alcfgJi) {xyzf'] = 



represents the general equation of a binomial qnartic curve having the 

 points 2 = 0, ^ = 0; s = 0, 2/ = for its two nodes. 



% {ax + /3y + yz) + xy - 



denotes a conic passing through the same two points. 



If it has double contact with the quartic, then it must also have 

 double contact with the conic 



(az + )8y + yzy + {ahcfgh) {xy%y = ; 

 hence 



(cue + /?«/ + yzf + {alefgh) {xy%f + 2A [z {ax + /8y + y%) + xy~\ = L^ ; 



and therefore every minor of the discriminating determinant must 

 vanish. 



Let A denote it. 



a + a^ h + X + a/S ^ + a (X + y) 



h + X + ^a b + fi^ /+yS(A + y) 



g + {X+y)a f+(3{X + y) c + (A + y)- - A^ 

 10 



a a ^ a- h -{■ X + a/3 ^ + a (A + y) 



13 h + X + /3a i + ^' /+^(X + y) 



A + y ^ + a(A + y) /+/5(A + y) c + (A + y)^ - A- 



1 - a - y8 -{^ + y) 



a a h + X g 



^ h+X b f 



y 9 f c-X" 



and every minor of the latter form of A must vanish. 



