CO VARIANTS OP PLANE CURVES. 
1185 
§ 2 . On the Conditions of a Covariant Function, and on the modes of Eduction 
and Development. 
7. The Form Conditions. 
The equations (A) (B) and (C) determine the form of the covariant functions; we 
will write them respectively 
(Oi-O,)/=0 (A) 
(O 3 '^ 2 )/ — ^ (®) 
(O 3 — Q.^f— 0 (C), 
where 
0 0 
Oi — (tt ]d) 0 ^^ 111 = ,iCp_iypU &c. 
Expanding the expressions for and Oo, we obtain 
* = 
O., = 
Ho = 
0 0 0 0 
^yi 0 “ + lOi/ 22/3 ^ + (15yoy^ + 10^3^) ^ 4- (21y,y5 + 2>by.,y^) 0— 
+ (282/3^0 + 56.%y5 + + &c. 
0 0 0 0 0 
%2^ + + 15?/^^+ 243/5^ + 353/8^ + &C. 
8^5 
8^/7 
% 2 %|^ + 30yo3/3 ^ 4 ( 603 / 23 /^ + 40 ^ 32 ) ^ 4 - (105yo3/5 4- 175^3^^ ~ 
+ 42 4- §^ 3^5 + 5y/) 4- &c. 
Relations hetiveen the Operators in the Form Conditions. 
It is easy to prove that we have the relations 
(Oi 111 ) (^2 (^2 112) (^1 Hi) — O3 H 
(O3 Hj) (O3 113) (O3 123) (Oj 122) = 0 
(O 3 - 03 ) ( 0 , - n,) - (Oi - nj (O 3 - 123 ) = 0 
(16). 
* The operators Qj and Cj are considered most conveniently in the form (25). 
Qj = 0 is the partial differential equation which is satisfied by invariants in the theory of forms, and 
= 0 was established for reciprocants by Professor Sylvester, in his lectures “ On the Theory of 
Reciprocants ” (‘ Amer. Jour. Math.,’ vol. 8, p. 238). 
I have not been able to consult a paper by Sophus Lie (‘ Mathem. Annalen.,’ vol. 32), in which I am 
informed that this operator is applied to kindred pui’poses .—August 2, 1893. 
MDCCCXCIII. — A. 7 M 
