137 
and similarly with other parameters for any number of consecutive points. On 
substituting these series expansions of x(i), y(i) in 9, we shall evidently obtain an 
invariant function. If now 9 be capable of expansion in a power-series with 
regard to dt, dr, ..., we shall have the coefficients, I, (x, y, x’, y’, ...), I, (x,y, 
x’, y’,...), ..-, of the powers of dt, dr, ... separately invariant, since the para- 
a, 
meters t, rf, «... are arbitrary. In I,, I,, I; .... we may express y’, y’’, y’””,... 
AR MUNCHONSIOL, yrnse Yast ete Kg Ky, Xp statins ma Livthen. ly 0 hss... mayaue 
so combined as to eliminate the differentials Xxx (ee wer shall obtain 
MivArlanin tUMCtLONS: Oy) (XS) Ys Va5 Yay ee ee ), $2) 93, ---., Which are differential 
invariants in the sense already defined. 
The calculation of differential invariants by the method just outlined is 
sometimes quite laborious. Below is given a consideration of some of the more 
characteristic groups. 
SECTION I. DIFFERENTIAL INVARIANTS DETERMINED BY TWO POINTS. 
In the present section are computed the differential invariants for some of 
the more simple groups of the plane, and indeed for such as have point-invariants 
for two distinct points. Only two differential invariants have been determined 
for each group; all others may be found from these by differentiation.* 
1. The group 
Lala | 
has the point-invariants x(i), ~, —y—y(2). Expressing y(2) in terms of a para- 
meter t, we have ultimately 
/ Ble eee ye 
vb, =y —(y i y dt eh i 2 “= area) 
Since dt is arbitrary, y’, y”, .... are singly invariant. 
y = x’y,, but x’ as well as x is invariant, hence y, is invariant, and 
our differential invariants may be written 
2. The group 
Pa | 
has the point-invariants 
u, =x — x(2), v, =y — y(2). 
Hence, we have 
ee dee ae dt? 
Hye (Xe he ee Vn yy — (yb y dt y ame 
a 
*Lie: Math. Annalen, Bd. XXXII, p. 220. 
