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X. 



A METHOB OF REDUCTION OF A QUARTIC SURFACE POS- 

 SESSING- A NODAL CONIC TO A CANONICAL FORM. 

 WITH AN APPLICATION OF THE SAME METHOD TO 

 THE REDUCTION OF A BINODAL QUARTIC CURVE 

 TO A CANONICAL FORM. 



By JOHN ERASER, M.A., F.T.C.D. 



Eead December 14, l!S(03. Published January 26, 1904. 



The equation of a quartic surface possessing a nodal conic may be 

 written in the form — 



[cur- + Py"^ + yz^f + w"^ {[_a, h, c, d, f, (/, h, I, m, n] [xyzivy] = 0. 



We may write x for x^y a, &c. ; and then the equation becomes 



[x^ + y^ + %^f + w- [ [_ahcdfgh Imn] [^xyzivf] - 0. 



If the quadric 



x^ -\- y"^ + z^ + [ax + l^y + yz + Sw^w = 



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

 with the quadric 



[ax + fiy + yz + Stay + [ahcdfgh Imn'] [^xyzw^ = ; 

 and hence 

 [ax + fSy + yz + Sivy + [ahcdfgh hnn~\ [xyzw^ 



+ 2X[x^ + y"^ ^- z- + w{ax + Py ■iryz + hvy] = LM, 



where Z = and M= are two planes. 



Hence, since every plane meets this quadric in a pair of lines, 

 every first minor of the discriminating determinant of this quadi'ic must 

 vanish. 



Let A denote this determinant : then 



a + 2X + a'' h + a/3 ff + "-7 I + a{X + 8) 



h + 13a i + 2A + y8- /+/?y m + l3{\ + B) 



y + ya f+j/^ c + 2X + y" fi + y{X + S) 



l+a{X + 8) m + /3{X + S) n + y{X + 8) c + {X+8y-X 



K2 



