136 
The group Xxf so extended ee: 
7 — Dee 
Mi a a 
dy 
* 
> © 
Lie has Mss that the extended lt i cai form an r— parame- 
oe Re) 
dy™) F 
ter group since the bracket relations 
(Xi. 3 Xy™) = —— Xf (X at) — XK ™ f) = — 
Seieee Re , ealsire eel 
; 
exist. But when relations (1) hold, the equations 
di ages te df 
ease os a ee Ss; (i) =— 
q vs ae a = T's dy® 
are known to form a complete system of linear partial differential equations in 
2--m variables. This system has at least 2--m-—r independent solutions 
which are defined as the differential invariants of the group Xxf. 
In Lie’s paper cited above it is shown that if two independent differential 
invariants be known, all others may be found by differentiation. For example, 
if the two fundamental differential invariants be ,, 0,, then 
p, do 
t= Ft b= Bt oles Tous 
The fundamental differential invariants 0, (x,y, y,, Yo, ----Yr—1), %» (x, 
Y, Yi, Y2,-- Yr), of an r-parameter group may, in general, be obtained from a 
somewhat different point of view, and indeed without a knowledge of the form of 
the group itself, provided the point-invariants of the group be known. 
Let us suppose the points of a point-invariant 0 (x, y, x/), y(!....) to lie 
upon a curve x = f, (t), y =f, (t), 
where f,, f, are analytic functions of the parameter t. Weseek the nature of the 
invariants when two or more points upon this curve approach coincidence. If 
x, y be a point for t= to, then a point x ®), y ™, ultimately coincident with x, y, 
will be given by 
2 2 
x(2) = x—+x’dt+ x Op Lon YOmy tye py” SM ee 
*Lie: Ueber Differentialgleichungen, die eine Gruppe gestatten. Mathematische 
Annalen, Bd. XXXII. 
+Throughout this paper we shall employ the following notation : 
(a) x,y; x(2), y(2); x(3), y(3); .... are points of the plane. 
,__ ax nf Loy SALy. Prat ch f 
(b) x eS res = de eee Vy ma | ae vo dt 
(ec) y= dy a vy, ; hence, we have y’ = y, x’ 
71 dx 72 ~~ dx? EAR, ’ ’ Ya ’ 
¥’ = yo (x)? + yx, Y= ¥o(x)? + By ex’ x” + ik «sss 
