DIFFERENTIAL INVARIANTS OF SPACE. 
3U) 
aX- + 2r/XZ + cZ'^ 
+ 2AXY + 2/YZ 
+ />Y“ 
aX" + 2g-XZ + cZa 
+ 2hXY + 2fYZ 
+ bY~ 
a"X’^ + 3 g-''X-'Z + . 3 c"XZ~ + ii"Z\ 
+ 3 h"X'Y + erXYZ + 3 m"YZ’ 
+ 3b"XY- + 3rT“Z 
+ k"Y3 
bug Let any such dilferential invariant 
be liomogeneons in of degree j); in a, h, c, /’ ;/, Ii, of degree q ; in 
a, b, c, f, g-, h of degree r ; in a", b", c", f", g", h", k", \", m", a'' of degree s ; then the 
index /a of the differential invariant is given hy 
3 /x p 'Iq d- Sr d“ th.s'. 
Further, a.s in § 18 , there are five e(|iiations out of tlie set wliich are satisfied hy 
the differential invariant when it is an invariant of tiie ternaiw systeni, and l)v tiie 
leading coefficient when the differential invariant is a contra variant of the ternary 
system. These equations are 
e,{t) = (), cq(0 = 0, 0,(0 = 0, c,(0 = <h C5 (0 ='h 
where the operators tq, Co, c,, tq, are 
' ^ a«. 
+ i> 
a 
ch 
+/¥ 
"N 
+ 2h 
: +b 
^ d- f 
•A 
ca 
ah 1 
eg 
d- 3h' 
/ a 
fa" 
+ 
2b" 
ah" 
+ k 
'V 
T fJ 
a« 
+ c- 
a 
da 
+ 2g 
+c 
f +f 
( 
ca 
eg 
ch 
+ 3g' 
. a 
aa" 
+ 
t, 
eg 
d- u' 
or ^ 
Co = 2 / - 
db 
+ ^ 
0 
¥ 
+ " k 
vh' 
eg' 
d-r 
d- m' 
f c 
o 
( 
d- 2f' ' d- m" ' d- V 
cc" ^ ah" ^ ar ^ ab" ’ 
, a . a . a 
+ ^hb + 'af 
d- 3r 
ak' 
+ 2m" ~ d- n" + 2f"^~ d- + g^'' 
ar 
am' 
ab' 
a_ 
a'f" 
ah'"^ ’ 
