COVARIANTS OF PLANE CURVES. 
From which we find the differential invariants of orders 7 and 8 conveniently in 
the forms 
Us = ~ SzigLgLy + 2L(3® . ( 31)5 
Ug = ^^5^9 ^LgLg -j- 3L.j," J 
which present’the same forms as the irreducible matrices of order 4, 5, and 6. 
16. The differential coefficients of the irreducible matrices, and of the quantities 
by which they are replaced. 
In this section of the paper the relations between the matrices, &c., are being 
investigated for the purpose of use in the next section. 
The differential coefficients of the irreducible matrices of even order are readily 
written down, and those of the matrices of odd orders with rather more difficulty. 
Thus 
du^ 
dx ~ 
d%n 
Uo -f = 
and 
dx 
dug, 
% = 
^ dx 
du 
5u^ + lOu^ag 
7Uq + 147 ^QCt 3 
Q d'tbrt 
ICcT — = 
— = SLg + AOufjQ + 907^5^ + l^u.^Uqa 
(32). 
Hence 
UM 
dLg 
dx 
" dx 
d\]g 
If 
■2^5 
UoU, 
— 7U7 + 7Lq^ — upi^ + ‘ZOufi^^a^ 
— 8Ug + I6U7L0 + 55 %^ + iQu - fJqa ^, 
= 9^58^9 - 21U/ - I2U8L9 - 54U7L93 
9L94 + eOWglTgag 
17. Expressions for dg, a^, &c., in terms of u^, Lg, differential invariants and a^. 
These expressions are found consecutively by the aid of the forrouiee which have 
just been established. Thus, 
MDCCCXCIII.—A. 7 N 
