PROCEEDINGS 



OF THE 



Camtritrg^ ^^tbsnp^kal Sfltbtg, 



Note on a property of orthogonal covariants of a binary 

 quantic. By Arthur Berry, M.A, King's College. 



[Received 17 March 1905.] 



Let f=a x n + na 1 x n ~ 1 y + ... + a n y n be a binary quantic, let 

 fl, O denote the usual operators 



d _ d d , v\ d 



a - n — h2a lT — ..., na^-j — V(n — l)a 2 -j— ..., 

 da^ da 2 da da x 



and let A be written for XI — 0. A function of the coefficients 

 and variables which is unaffected, or changed in sign, when the 

 variables undergo an orthogonal transformation is called an ortho- 

 gonal covariant. I consider in this note only such " direct " 

 orthogonal covariants as are unchanged by orthogonal trans- 

 formations of the direct or rotational form 



x = cos 6X — sin 6Y, y = — sin 6X + cos 6Y. 



It is clearly sufficient to consider only such covariants as are 

 homogeneous both in the variables and in the coefficients. The 

 necessary and sufficient condition that a rational integral function 

 of a , a x , ... x, y should be a direct orthogonal covariant is known* 

 to be that it should be annihilated by the operator 



V A_ X A__ A (1) 



J dx dy v ! ' 



Let the covariant be 



G x m + G x x m ~ Y y + ... + G m y m = 0. 



* See for example Elliott's Algebra of Qualities, chap, xv., or Sylvester's original 

 paper quoted below. 



VOL. XIII. PT. II. 5 



