452 MR. A. R. FORSYTE ON INVARIANTS, COVARTANTS, AND QUOTIENT- 



for the transformed, and therefore 



Va _ .Vs _ y* 



y\ y* y 



for the untransformed, equations hold. (It should be remarked that these are not the 

 most general pair of quadratic relations ; in fact, interpreted geometrically, they 

 represent a pair of quadrics which by their intersection determine a tortuous cubic.) 



We now proceed to prove the converse that, if two quadratic relations of the 

 foregoing type hold, then the priminvariants vanish. 



Taking the four solutions of the equation in the form 



M! = A-', u. 2 = XA-, u 3 = O-A-', M 4 = />A- J , 



the relations given are equivalent to the new relations 



o- = X 2 , /> = X 3 . 

 When these values are substituted in A, it becomes 



A= X" 1 , X", X' 



2XX m + GX'X 11 , 2XX" + 2X' 2 , 2XX 1 



3X 2 X m + ISXX'X" + GX' 3 , 3X 2 X U + 6XX' 2 , 3X 2 X' 

 = - 12X' 6 . 



Since any constant factor may be absorbed into the particular solutions u lt u%, Wg, u^, 

 we may take 



! = X'-. 

 Again we have 



= MI (X 1T + 4Q 3 X') + 4' 1 X m + Gw^X" + 4w lil 1 X i , 

 = M! (a** + 4Q 3 o-') + 4tt 1 1 o jii + GM^CT" + 4M ili 1 o J . 



When in the latter we substitute cr = X 2 , and from the resulting equation we 

 subtract the former, multiplied by 2X, the new equation is 



= Ul (8\ l \ m + 6X" 2 ) + u\ . GX'X" + 6u\ . 2X' 2 , 



which, by the substitution of the value of u lt changes to 



= - 10X' (X, z], 



or, since X' is not zero, we have 



(X, z} = 0. 



We therefore take X = z ; the four solutions become 1, z, z 2 , z 3 ; hence Q 3 and Q 4 are 

 both zero, and the priminvariants vanish. 



