296 
riiOFESSOR A. E. FOESYTH ON THE 
c cr 
CM 
I 7 ^ ^ I ^ r (X I -x-i C (X t V C'CX • A C (X 1 f 
+ + b + f + </., 
rcr 
c;a 
ch 
eg 
010 - 
^'*^100 
0. 
^ r;cr 
CH 
, LU , OCT , , OCT . r ( cr , 
+ / o, + c ^^-+2g'_-+ f + 
eh c(j ca eh 
( (7 I . f cr 
c + 9 u017-,— 
^9100 
rcr 
= 0 , 
rcr 
ecr , rcr rcr , rcr . , rcr , ^ r , , 
" r/, + a + ;/?/+" a + ^ rf + Cf..,,., 
= 0. 
,ccr , ,.’rcr rcr , 
!/-j .r + C „ -f 
c/i' ch rf 
car 
'h 
+ 
+ 
cb 
9? . ^ 
''.p I 9ooi / 
ct C^oiO 
= 0 , 
C'Cr 
, I 7 ecr . L'o , 
r,, + V,- + -"//— + 
(7/ f;/ (:C 
a 
eo" I 1 
eg 
ca- 
df 
+ 
^ ecr . . 
-g-. + </> 
(C 
rcr 
lOu 
C(f)Q 
= 0 . 
001 
7 rcr . T uu . c)j:‘Uij , , rcr , , rcr . ,~,^rcr , , 
h 4- b ^ + 2y — + h ^ + b + 2f ^ - + (/)uio . , 
dj- OC eg rf re cc^, 
eJer 
da- 
c cr 
OOl 
= 0 , 
co¬ 
ca- 
olr,^'^ 1, ^cr\ . rcr „ccr . j rcr , 
-'"S.S6j+^'s. -h a?+- (" aaa )+8 “Or+^“a-^- 
a 
rcr rcT' 
acr_ ,£cr , .^/ ?cr_ acr^ 
aa “0c/ ■ ■ dh ‘^0/-^‘^r0a 0c 
)+/^ 
ag- ef 
i+>*ah-4f+^ 
Alo 
C(/)q 
-0, 
rcr 
'0> ^ 
J00:jr *^001 
100 
The ninth equation takes tlie foiiii 
rcr 
^'^001 
= 0. 
V(T 
o / I 7 dcr , rcr . 0cr , rcr , , 
2 « a,; + '> sc- + <•' 3r + / s>- + y o- + 
0a 
dh 
0 c 
+ 8(ar +b^^ + cr--f-f 
\ ca ob re 
cor 
Wi 
+ I 1 0 cr 
^ag + ‘‘ah 
rcr 
I , rcr , j, rcr . , rcr 
+ 9iou VI-r 9oio WT-r 9ooi VT— 
^9010 e9ooi 
oficr. 
Insjoection of the first eight equations in this form, coupled with the limitation 
that some at least of the quantities a, b, c, f, g, h occur, shews that they are the 
difterentlal equations of those invariants and contravariants of the simultaneous 
ternary cpiadratics 
(a, h, c,f, cj, /.XX, Y, Z)\ 
(a, b, c, f, g, hXX, Y, Z)g 
Avhich involve the coefficients of the second of these (piadraties, and that the contra- 
gradient variables (when they occur) are c/>,iiq, ^,, 07 . The ninth equation is 
satisfied when the index /a is appropriately determined as folloAvs : let cr lie homo¬ 
geneous in <^Q 7 Q, of degree n, in ci, h, c, f, r/, h of degree m, in a, b, c, f, g, h 
of degree I ; then 
Si -j“ 2in “I” — 3g' 
