122 
This is the only group of the class (A), and furnishes us when eztended the 
linear partial differential equation 
n 
VW/ Lou 
he dyi 
The invariants of the co-ordinates of n points by this group will be the 2n —1 
independent solutions of Wf—0, i. e. 
Xe —— Ve lh — le ge 1] - 
B. All families of curves of the form ax +- by — constant remain invariant. 
oe Nia 3: as 
The complete system corresponding to this group is 
Wi ate 5 = _ Seg dt 
1 dyi 
=—i()5 
with solutions ; 
$j =x, — Xj, i= yi. — yi, (J=2.... n). 
The functions ¢, / are the required invariants. 
3. | q,xp+yq |. 
From this group we have 
r >. di pees dt df 
ee ee eee eae 
Ww, S dyi ’ of } 1 et ngtl 
These two linear partial differential equations evidently have as solutions 
_ yi—yk,,_ yi— 2, G25 se ore UL) 
J Xy yl—y2 Xl 
Ge. 
which are the invariants sought. 
4. | p,q, xpt yx |. 
This three-parameter group furnishes us the complete system 
n n nD 
dt =. at = di dt 
>i =— = di — = + —— +> =—0. 
Gan = dyi Bi [ag eg dy 4 
The first two of these equations have solutions 
= x1— Xj, i= y1— yi, (j= 2 .. n), 
which as new variables reduce the last equation to the form 
n 
Sele, Mle aay well dO 
ma ) 1 a0, Vi aij 
Hence, the invariants are 
Ti cee tee ee a eee 
d2  x1— xe’ we -yi— yea’ X1 — x2 
