426 Differential Covariants of Plane Curves, fyc. [May 4, 



and its position depends npon the invariant ratios \ : fi : v. Treating- 

 (X : fi : v) as determining the position of a point, and (0 X : 02 : 03) the 

 position of a line, the condition that (X : /i : v) may lie on (0j : 2 : 3 ) 

 is 



^0i + yt*02 + ^03 = 0. 



Hence they form a correlative system of point and lire coordinates. 

 I define (\ : /* : v) as the invariantal coordinates of a point of homo- 

 graphic persistence, and (0i : 02 : 03) as the invariantal coordinates of 

 a co variant line. 



The condition that (X : jx : v) may lie upon a co variant carve of the 

 n th order will be an invariantal relation, homogeneous of the n ih 

 degree in A, v, between A, v and the invariants in the coefficients 

 of the equation to the curve (we may say the invariantal coordinates) 

 of the curve. 



If f(\ : ju, : v) = expresses this relation, it is in this system of co- 

 ordinates the equation to the curve ; I call it the intrinsic invariantal 

 equation to the curve. 



The coordinates of the tangent to the curve at (X, : /x : v) are 



and (v jL+^A+v' ^-)f= is the equation to the 



\oX O/i Cv/ \ OX Of*- Ovl 



first polar of (X' : ju! ;v) with respect to the curve. 



Also writing x for fijv and y for XJv, the relations between tt, % and 

 x, y are of the form of a homographic transformation, and therefore 

 any function of <2V/cjjp, &c, which is a differential invariant, is equal 

 to the identical function of d 2 y/dx 2 , &c, affected by a factor of known 

 form. Hence, treating f(X : fi : v) = or f(x, y) = as an ordinary 

 algebraic equation, it will have the same homographic singularities as 

 the original covariant function, while the coefficients are the differ- 

 ential invariants which characterise the curve. 



The intrinsic invariant equation to the osculating conic is 



\i/ + /r = 0, or y + x 2 = 0, 



and to the non-singular osculating cubic is 



ufX- (V 8 \ + U» + (U 7 2 \ . - V 8 jn + TV) O + /* 2 ) = 0, 



To terminate the abstract, the equation to the polar conic of the 

 origin is 



U 7 2 \ 2 — V 8 V + U 7 ^ 2 + 2U> = 0, 



and therefore U 7 2 \— Ysfi+U-,* = is the equation to the common 

 chord of this conic and the osculating conic, and it touches the cubic 

 at X == 0. fi : v = U 7 : Y 8 , the tangential of the origin. Also the 

 second tangential of the origin lies on 



Y 8 \ + U 7/ i = 0, 



