newson: linear geometry. 87 



2 



For k=o, we have p^+wH^o; or Dp''^-\a.^(\vp^ — w2H)=o; 



2 J. 



For k = i, we have p^-(-^v2H=;o; or Dp3-^-aj(w2p — wH):=o. 



It will be observed that the first factors in each of the expressions 



above are factors of the expression p"-|-H''=o; and the last factors 



2 3 



are factors of the expression D''^p-f3D2pa H^-a pG=o. 



It can readily be verified that p3-[-H3= — p^ (G2+4H'^). Since 

 ^ ((j2-j-4H3)=a'-^, where D' is the discriminant of the cubic; we 

 have :i-|)-^=(p2_|_H3)2. Hence p"^ -{-H^=-o is equivalent to 

 ^^-o. ^, is a fundamental invariant of the cubic only. 



It can also be readily verified that 



3 



p(D3J 3D2a'H-fa G)=— a'-^pE 



3 



where E=ab — ^ba.b'-i- :;ca''b, — da , the eliminant of P and C, 



Thus, by giving to k the critical values r, o, CX), we obtain two fun- 

 damental invariants whose vanishing expresses in the one case the 

 condition that the point P coincides with one of the points of C; and 

 in the other, that two points of C coincide. 



If in equation (i) we give to k the values — w and — w-, we obtain 

 after reduction the following values: 



When k= — w, we have p3(DH-|-ap)=o, 

 when k= — w^, we have pD — a^H^^o. 



These expressions equated to zero yield, after squaring, the same 

 result. Their product is free from radicals and gives us 



p3(D-H — ajDG — a'H^). It can readily be verified that 



D2H— a,DG— a'H2=a2E,, 



1 1 1' 



o 2 



where Ej=(ac — b-)b'' + (bc — ad)ajb^-f-(ad — c")a . 

 We recognize this at once as the eliminant of P and the Hessian of 

 C. This is another fundamental invariant of the system of point 

 and cubic. 



I 

 Again let us give to k in succession the values — , 2, and — i. We 



2 



have after reduction the following results: 



When k= — then (Dp- a, H^)— p^(ap + DH)=o; 



2 



1 

 when k^ -2 then (Dp — a^H^) — wp'^(ajP + DH)=o; 



when kT^. — I then (Dp— a, H2)— w2p^(a,p-Ll)H)=o. 

 F^ach of these expressions after expanding and cubing leads to the 

 same result. Hence we may more simply use their product which is 

 (Dp— aiH3)3— p(a^p + DH)3^o. 



