148 
11. By the group 
q, YG, Yq, P, xp 
the functions I,, I,, I, of 9 remain invariant, also —, —, —, .... as in’8. 
> CI AR 
These invariant functions must be so manipulated that the x’s are either elimi- 
4/ x’ 
nated or made to appear as ratios Sh eae tack Since: i (x), I5Ge) . are 
tf SI, 
F x? 1x : PRN 
already functions of a7) gf? we may omit these, and write simply 
t)38 + ¢, 0 ee 
J, =46,(x’)’, J 
L / Boe 
J, = $4 + a(x) "x" + 4? Sone 
Eliminating x’, x ; 
Yaues SVawWieens (alle 
( 5 fe ae a Ps Yi Vie ram Neisite oa 
{Je 1 7} iI y= (0,)3 lee aa 1 
tn) 
te DP (ee (fe me 
poe by Ga ard, ! © go. fax? 
ob, (%’) ( SO Gaara Jr 5 le 
Rts so Oe sae oo” ox 
Hence, A =— (x’)? is invariant. 
2 s { y. 2 a2 3 4 
da ig Maas Ea 417327 ¥a_g evze) 
pie ete PU ga Yi Wat) Yi Yai) see og 
cron ae aa 2 
vf aot J 
®,, ®, are the two fundamental differential invariants. 
12. It has been shown that the group 
DS Gig) Nem... Aokrg, 7). Xiang 
1c cal | 
leaves invariant x and the determinant 
Nf ye ye eh Seale 
ENG; Gey ee ee) het) YOK Ge tD) 
| X@ XG BGO of KGET D) 
