1194 
MR. R. ¥. G-WYTHER ON THE DIFFERENTIAL 
Clc) 
Cto — 
a, = 
05 
«6 ■-= 
W 3 
«3 
«g + 3«4«3 + a.^ 
L 3 -j- 4 ” “ 1 “ cu^ 
a. = 
_ Uy + L/ + 5u^it.- + Swg (Lg + 2?f/) ^3 + + 10 + a^ 
(33). 
Oq = 
Ug + -f- 3UyLg + L^g + 6t^^?(j''Lg + (U^ + Lg^ + ci,^ + &c. 
( lc\ — 
- 3U 2 + &c. 
In these formulse, and in their applications, it must be remembered that they imply 
that U .2 and are not zero at the point to which they refer. 
§ 3. Correlative Forms. 
18 . IVie deduction of the equation to the reciprocal of a covariant from the equation 
to the original covariant, 
[The jDrinciple of duality, as applied to differential invariants, is explained by 
Halphen (p. 56) in his thesis. In the case of covariants, the relation is still more 
interesting. 
The general investigation will come later, but let us begin by considering the 
connection in its most elementary form. Let a?, y, be the coordinates on a covariant 
in either point- or line-coordinates, and X, Y, the corresponding coordinates on 
the reciprocal in line- or point-coordinates. We express the correlative transforma¬ 
tion by 
X = 2/i 1 Ja; = Yi 
Y = mj,-y\ ^ ly = XY,-Y 
yi and Y^ being understood to stand for dyjdx and dYjdY. Then we easily find 
Y, = -, Y3 = 
2/3 
*3 
y; 
3 ’ 
J 
vdth the inverse relationships. 
These equations enable us to express any function of Yg, Yg, Y 4 , &c., as an equi- 
