COYARIANTS OB’ PLANE CURVES. 
1209 
If three points (X : p,: v), (X': /r': v'), (X" : fx" : v') lie on a straight line 
X, IX, V 
K, IX, IX 
// 
V 
a,nd so on. 
I define : [x w) as the invariantal coordinates of the point,, and {(j)i : </>2 : <^> 3 ) as the 
invariantal coordinates of the line. 
The condition that {x :v) may lie upon a covariant curve of the order v^ill be 
an invariantal relation between X, fx, v and the coordinates of the curve, homo¬ 
geneous of the degree in X, [x, v. 
It follows that if /"(X, jx, v) = 0 expresses this relation, it is, in this system of 
coordinates, the equation of the curve. I shall call it the intrinsic invariant equation 
to the curve. 
The coordinates of the tangent to the curve at {k: jx :v) are 
0 / _ a/ . 
d\ ' dfx ' dv ' 
If (X' : p,' : v') lies on the tangent to the curve at jx :v) then 
w3/ ,0/ '^/_n 
^ 3,. +" a. - 
Taken with f (X, fx, v) = 0 , this gives the coordinates of the points of contact of 
tangents from (X': fx' : v), and, being of the n — 1“" degree, it represents the first 
polar of f (X, [X, v) = 0 at (X' : [x' ; v'). 
Finding the first polar of this again, the second polar of /(X, [x, v) = 0 has the 
equation 
+ 
IfX 
0 
It is unnecessary to carry this further to determine how far the ordinary methods 
of geometry apply to this novel system of coordinates, since the character of the 
analogy is obvious. 
31. The values of the di fferential invariants at different points on the curve. 
The coordinates of any jjoint on a covariant curve were, (55) and (56), given in the 
form 
TT = — ufufY j — {u^uf Ijf -b u-ffpifj Y + (Lg -f upifj X -|- 1, 
^ — WgWg (LgY — X) / — {u^uf + Lg^ -j- u-fjpt^ Y -b (Lg + upi^ X -f- 1, 
MDCCCXCIII.—A. 7 p 
