of orthogonal covariants of a binary quantic. 57 



Thus <f> m = (A 2 + m 2 ) (f> m _ 2 (A). 



Now (/>(1) = A 2 + 1, <j> (2) = (A 2 + 2 2 ) A, 



so that, if to is even 



<f> (A) = ( A 2 + m 2 ) {A 2 + (to - 2) 2 } ...(A 2 + 2 2 )A (3), 



and if to is odd 



(/> (A) = (A 2 + m 2 ) {A 2 + (m - 2) 2 } ... (A 2 + l 2 ) (4). 



The equations (I) shew that, if the source G is found so as to 

 satisfy the differential equation (2) then G 1} G 2 , ... and hence the 

 whole covariant are determined uniquely. If the source of a 

 covariant of order to is annihilated by an operator ^> m _ 2r (A) 

 obtained by omitting r factors at the beginning of $ m (A), it is 

 at once evident that the covariant is the product of a covariant 

 of order to — 2r by (x 2 + y 2 ) r . It is suggested that, if the source 

 of a covariant of order to is annihilated by an operator made 

 up of some other selection of the factors of cf) m (A), then the 

 covariant is expressible in terms of covariants of lower order, but 

 I have not succeeded in proving any such result. 



The result of this note was given in a question set in the 

 Mathematical Tripos, Part II. 1897. I had not then read 

 Sylvester's original paper " On the Principles of the Calculus of 

 Forms*"; I now find that the equation 0(/e) = O occurs there 

 (p. 359 of the reprint) as an auxiliary equation in connection with 

 the problem of integrating the differential equation corresponding 

 to (1), but its significance as a differential equation satisfied by 

 the source is not indicated. 



* Cambridge and Dublin Mathematical Journal, vn. (1852), pp. 179 — 217, re- 

 printed in Mathematical Papers, Vol. i., 43, pp. 328 — 363. 



5—2 



