320 
PEOFESSOK A. P. FUK^YTH OX THE 
Kt 
( c 
+ 2f: + b ’ +h 
( c cl 
+ rn" + rr + rc'' + rr' + "" ■ 
= 2/,- 2.; F + /A. - y F 
cl) (c oh ((/ 
+ 2b ' - 2c ! + li 4 
cc 
Ti 
g- 
(g- 
o , n// B /, B 
+r ,-m 
( K (1 
r Yii" c n 
cb'- cc" ' <'h" ° ?g" 
These, eijuatioiis involve 22 arguiuent.s; being themselves 5 in iiiiinhei- and a 
complete set, they ])ossess 17 solutions. 
A<jgrc(j<ite of lavarlauts of the Third Orda-. 
2 G. The asyzygetic aggregate of concomitants of two ternary quadratics and one 
ternary cubic has not yet been constructed, so far as 1 am aware; and it therefore is 
not 2)ossible to select from it an algebraically complete aggregate of invariants and 
contra variants. But tlie established knowledge of the asyzygetic svsteni of two 
ternary quadratics''^ and of tlie asyzygetic system of the ternary cubicf is sufficient to 
]>ermit the construction ot this algebraically complete aggregate; it is therefore 
unnecessary to pi'oceed with the formal solution of the preceding live ecjuations. 
The most compendious way of ex'pressing the results is to use the cirstomarv 
symholical notation. \\ e use the umbral symhols adopted in ^ 17 ; and, in addition, 
in connection with the ternary cuhic, we introduce umbral symbols a.,, a .^; /S.,. ; 
and so on. Then in the usual notation, we can write 
{<(, 1), r, f p, hJX, Y, Z)- = af = l)f = ... , 
(a, b. c, f, g, hXX, Y, Z)- = nV = h'd = _ 
(aY b", c", f", g", b", k", 1", m", n"XX, Y, Z)' = «/■ = /3f = .... 
1 hen an algebraically complete aggregate ot invariants and contra variants of the 
two quadiatics and the cubic is constituted bv the following seventeen members; 
* See the reference in § 16. 
t Goruax, ‘Math. Ann.,’ vol. 1 (1869), pp. 90-12S; Gr.NDELriXGEi:, ‘ (Math. Ann.,’ vol. 4 (187P, 
}ip. 144 163 ; Cayeev, ‘Cull. Math. Papers,’vol. 11, pp. 343-356. 
