125 
We may choose the solutions of this last equation as the cross-ratios 
ul (1 —us) «x2. — x1 _ xX, — XI i IN A A Ere _yl—yl 
we =: . Se i eee Ahoy, 
ae 3 (1 — w1) eke) ok xs vs (1 — vi) 2S yal = Ve 
(a) 
and the ratios 
Ot (le =a) a = WD 
== — . ’ 
w2 (1 — a1) Wil = VO SS yh 
Ue) (1 = Wa) Oo 
i == : j 
Weil == Wey) 39 SS as 
} aus. (lor). ae ks a — Xe 
~ @1(l1—us) xe—x3 yi— x2 
SECTION II. INVARIANTS OF SUCH IMPRIMITIVE GROUPS AS LEAVE UNCHANGED 
ONE FAMILY OF 0’ CURVES. 
The remaining groups of the imprimitive type leave invariant one family of 
o’ curves, and have been reduced by Lie* to such canonical forms that this 
invariant family is x = const. 
14. >. hee. Cc ae. 0 Pa Xrq 
In this group Xx is a function of x alone, and r>1. Each curve of the 
family x —const. remains singly invariant. 
The complete system of linear partial differential equations 
Wy f= Si Xy(x). 4 <0, (k= 
k Wits So es ) ( = seu Es 
corresponding to this group, has as solutions x1, x2, ... xn and n—r other 
independent functions Ds, (s =1, 2... n —r), which we shall define as the n — r 
determinants of the matrix 
ine al va oer nat DA (M,) 
e7 (ele! wp lefin, io ee) e <3 
“Lie; Math. Annalen, Bd. XVI; Contin. Gruppen, Chap. 13. 
