135 
il + wm ead restoring the variables 
Selecting as invariants Qm and Hm = aa 
Xi yi, we have 
2 mal tS mn | [124° | 23-4 
ae ira les aa ey oe Pee a | 
The forms of Q and H show that the general projective group leaves invariant 
the cross-ratios of five points. 
DIFFERENTIAL INVARIANTS DERIVED FROM Pornt-INVARIANTS. 
By Davin A. RotTHrock. 
In an accompanying article concerning Point-Invariants, the writer has 
shown how a group 
s lf di 
Xf = &(x, y) 5+ mx, y) > (K=1.... 0), 
may be eztended to include the increments of the codrdinates of n points. 
members of a group may be eztended in a different manner, and indeed so as to 
The 
include the increments of 
dy, d’y dy 
dx dx? dx? 
For example, the group Xxf gives to x and y the increments 
= &y ot" oy =r ot" 
=) COUN) Ss the increment 
— ) 2a] Wl 
Bye dx . ddy aes . Odx ie Aix — yds Bf i. 
dx? dx 
ais d?y ; : 
Similarly, y’” = —- receives the increment 
dx 
a 7/7 ia 
dy” = i ea ALTE eae i Ot, 
dx 
and.in general 
ss (Geni) an 
dy™) — dn dx aes Ot = mn ™ bt. 
