121 
The transformations of the 7-parameter group Xxf may be extended according 
to the method of (2) above, giving r 
W,f == SiXOF, (ie—L7 2: E); 
1 
which determine the increments of a function f (x,, y,; X2, Yo; .... Xn, Yn). 
Since the relations (3) exist for Xif, Xxf, they must also exist for Wif, Wuxi, that is 
W,(W,f) — W,( W;f) —— >; Ciks W =. 
ik 
Hence, W,f{—0, W,f—0, ... Wrf—0 are known to form a complete system 
of linear partial differential equations in 2n variables xi, yi, with at least 2n —r 
independent solutions. These 2n —rsolutions are the invariants of the co-ordi- 
nates of n points by the 7-parameter group Xxf. These solutions we shall call 
pownt-invariants. 
According to the method here outlined we shall determine the point-invariants 
of the finite continuous groups of the plane. In Lie’s classification these groups 
are divided into two classes: (1) Inprimitive, or those groups which leave 
invariant one or more families of «’ curves; (2) Primitive, or those groups leay- 
ing invariant no family of »” curves. Subdivisions of the imprimitive groups will 
be indicated in the text. 
Nore.—The results of the present paper were worked out early in the spring of 1898. 
Since that time there has appeared a short article by Dr. Lovette, June number, 1898, of 
Annals of Mathematics, upon the same subject. Only a few of the projective groups are 
considered, however. Among these are the special linear, and general linear groups. 
SECTION I. INVARIANTS OF SUCH IMPRIMITIVE GROUPS AS LEAVE UNCHANGED 
MORE THAN ONE FAMILY OF &’ CURVES. 
The groups of this category have been reduced by Lie to such canonical 
forms that they leave invariant: 
(A) @” of families of ,,’ curves: @ (x) -+-(y)—constant, 
(B) A single infinity of families of ,’ curves: ax + by —constant, 
(C) Two families of ,.’ curves: x = constant, y—constant. 
(A) The totality of curves $ (x) + (y) =constant remains invariant. 
lg q é 
* Lie employs this symbol to enclose the members of a continuous group; 
he dif 
== i= ; 
p dx? 1 dy 
