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V. 



ON THE PKOPEETIES OF A SYSTEM OF TEENAPtY QUADPJCS 

 WHICH YIELD OPEEATOES WHICH ANNIHILATE A 

 TEENAEY CUBIC. 



By H. G. DAWSON, M.A. 



Kead May 13. Ordered for Publication May 15. Published December 27, 1907. 



Section I. 



Taking the canonical form of a cubic curve x^ + jf + z^+ 6mxi/z, we have 

 two cubic contravariants, the Cayleyan 



F ^ m (a^ + j3H 7^ + (1 - ^m?) ajSy, 

 and the contravariant 



^ ^ (1 - lOm^) (a^ + j3' + 7^) _ ^^2 (30 + 24m^) a/By. 



If we take then the contravariant - 6 TB + 8>S'§, where B, T are the two 

 invariants of the cubic, we obtain the contravariant 



- &TP+^8Q - h \m {a' + ^' + f) - Sa^y} - D, 



where h = 2{l + 8m^y, which has an interesting connexion with the cubic. 

 The polar system of this contravariant is 



a (ma'' - /3y) + /3' {m(5~ - ya) + y' (my^ - "i3) ; 

 and we notice that if the symbols a, j3, y be replaced by differential symbols 



d cl d 

 dx dy dz ' 



respectively, the operator obtained annihilates the cubic form 



x^ + y'^ + z^ -{- Qfiixyz, 

 with which we started. 



It will be noticed that the property is independent of any linear 



transformation. 



The system of conies 



p {mx^ - yz) + q {my'' -zx) + r {mz^ - xy), 



which I call a system of annihilating conies, can be arrived at in another 

 very interesting manner, which shows their relation to the cubic in a fresh 

 light. 



