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VIII. 



METHOD OP OETAim:N"G THE CIJBIC CTJEYE HAYING 

 THEEE GIYEN COPTICS AS POLAE CONICS. 



By J. P. JOHNSTOIf, Sc.D. 



Eead Juxe 22, 1903. 



The problem to obtain the cubic which, will have thi'ee giyen conies 

 as polar conies was solved by Dr. Salmon {Conies, § 389 c) by finding 

 the equation of the cubic when the equations of each of the conies 

 was written in the form 



axP' 4- hy"' + c%^ + dw- - 0. 



The form in which he obtained the cubic in this case enabled him to 

 state at once that in the general case the cubic was the Hessian of the 

 Jacobian of the three conies minus twice the Jacobian multiplied by 

 the invariant T. The following is a different method of investigating 

 the same problem, by which it is seen at once that the solution, where 

 possible, is unique. 



Let the equations of the conies be 



u ^ (fl, h, c, /, y, hjxyzj = 0. 



V ^ (a',b',c',f',y',h'Xxyzf = 0, 



iv^{a",h",c",f",y",h"Xxyzy = 0. 



If these transform to 



Id^ 



^^SdX^ 



T^ 1 ^?* ^ 

 Y = IF- = 



3^F' 



Id^ 

 3d2r 



by means of substitution 







X = 



XX+yxF+i/^, 





y = 



\'X+/F+v'^, 





z = 



A."X + />i"F+v"^, 





then since 







dV dW 



dW dU dU 



dV 



dZ~ dY' 



dX~ dZ' dY~ 



dX 



