290 
PKUFESSOK A. R. FORSYTH OX THE 
and from the e([uatioiis (IL), (Hg), (Ilg), we have 
((T (rr _ Q ^ her _j_ ccr _ her _j_ her 
CKtoio 9^ho0 ^^100 ^t/oio 
•lOO 
Too 
‘010 
(cr I ccr _ _ .) ((T . ccr _ ( cr . ccr 
“T V - — o — w ^ ^ — -j- 
^^‘loo ^ 9 ooi h/ioo 
^ ^ 
ccr , ccr 
ch 
001 
OV + =0 = 2-'— + 
r/> ' hf he ' cf 
'-^01 VOlO *"*^010 VC 
■"010 
001 
ccr . ccr 
--r ^^T— 
^ifoio ^'^001 
These equations show that no one of the first derivatives of «, b, c, f, g, h occur in 
the hypothetical invariant; that is, there is no proper differential invariant of the 
first order in the quantities a, h, c, f, g, h alone. 
10. Next, are there any differential invariants, which involve a, b, c, f, g, h but no 
derivatives of these quantities and which also involve derivatives of a single function 
<f) of the first order but none of higher orders ? 
The nine characteristic equations are 
,y I ^cr , T her t ccr , , 9 , 
a« + ^ ih + -f 3ff + = «’ 
TOO 
ca 
,‘ha . _ ?<T , , 
J V, + YT: + voui 
C 
cli hg ' 
= 0 , 
cor 
GO" 
0O",^70O" , L-A7 , I 
“a/.+ + 
= 0 , 
ro- I .w.9cr I 9o- . , ccr ,, 
'Jdh + Y a;; +»+'#'0.1 - 0. 
hb 
9cr , 7 9cr . 8cr . , ccr 
+ ''' N/ 
hg 9/ 9c c(^oui 
o 9 
2 a 
cr 
9a 
crt ^ hg 
>‘%+^ a/ +^-^3; + 
icr 
> 9 cr 
ccr 
^ cr . I 9cr . . 
cj hb 90100 
4^010 ^ 
ccr 
h4>oui 
her 
^'4*010 
= 0 
= 0 
:a 
car 
ha 
\ i ^ 
.-9 
cr 
3/ 
2<.- .^ + <#■ 
cc 
co¬ 
rn 
90 
0Onl = 0 
100 
9^001 
a 
ccr 
ha 
, 7 9cr , 9cr I ^ccr , ccr . 7 9cr\ 
9c 
+ ^101 
90" I a 9cr , , 
+■ TOlO ATT-T 0OOL 
'Y 
ccr 
'4> 
100 
i0 
010 
90, 
= o/xcr 
001 
