THE EEV. GEOEGE SALMON ON QHATEENAEY CUBICS. 
237 
before mentioned. It is 
ahcdel{a j3){(a+ h)G^d‘^e^ — ahcde{cd -\-de-\-ec)) . 
The coefficient of a in this is 
dbcde {A — b¥(fd^e^ + 'bdbcdeihcd -\-hce-\-lcd-\- cde ) ) , 
A being the simplest invariant. 
The next linear contravariant is of the 21st degree, 
a^5V(ZV2(«-5)(a-i3). 
The next is of the 29th degree, 
a^V&d^e^'%cde{a--b){a—^\ 
and so on. 
The simplest covariant quadric is of the 6th degree, 
abcde{ay? -f b’f + cz^ + did‘ + ew®) . 
The next of the 14th degree, 
cC-b-&d'^e^'lab{cd + de-\-ec)xy, 
which may also be written 
a^b‘^(fd^e^^{bcde — abed — abce—abde — acde)x^. 
The next of the 22nd degree, may most simply be written 
a^b*c*d^e\a^a^ + + cV + d’^u ^ + 
That of the 30th is 
and so on. 
The contravariant quadrics are — 
10th order; 
18th order; 
The covariant cubics are — 
9th order; 
17th order; 
a^¥c^d^e^'Zcde{a— &c. 
'Xcde{a-\-b)abcde zuv 
a^¥c^d^e^{ + c^z^ + d^u^ + e%^). 
This last covariant is important, because we can immediately deduce from it, that the 
operation 
performed on any invariant or covariant of the cubic 
ax^ + + cz^ + du^ + ev^ 
gives rise to a new invariant or covariant of the cubic, and of the degree sixteen higher in 
the coefficients. It is on this account that I think it enough to write down two cova- 
riants of each degree in the variables. 
