1192 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
and 
Cl^u, = 0 
= u.^a^ 
= 0 
1 
= — Wg 10i^4«3 f* • • (29). 
Ct^Urj — — 2u^Uq + lOi^/ + 7u.a^ 
= — 3^7 + 21wga3 
= — 4 w 3%8 + 52 ^^ 3 ®^^ 4 '^<g — 30 {u-^ + 4 «/) + 
15 . Reduction of the matrices to functions of differenticd invariants, amd two 
fundamental matrices of orders 4 and 6 . 
On the reduction which is now introduced depends the possibility of making the 
subject of this paper an instrument of research into the character of the higher 
curves. 
For every order higher than the third, there is either a possible matrix or a 
differential invariant, and for every order higher than the sixth there is a differential 
invariant. Of the third order there is neither, and of the fourth and sixth orders 
there is no differential invariant. Every function of differential coefficients can be 
expressed as a function of a^, matrices of orders 4 and 6 , and differential invariants, 
and if the function is itself a matrix it will not contain 0 ^ 3 . 
As a first step, I replace the irreducible matrices of the sixth and higher orders, by 
matrices of homogeneous covariants, that is, by functions of the irreducible matrices 
which satisfy £l^f ~ 0 . 
As far as the 9^'^ order these are the irreducible solutions of 
and are 
du^ du^ die- dUf. dur, du^ dii^ 
0 uf 0 7n^ 21u^ 
Lg = ufuQ — buf 
E 7 — ““ 7 U^^U/^ 
Lg = ufu^ — 2\ufu^Uf^ + 70uf 
Lg = vfu^ — IQufu^Ur; — 2 >Qufu-^UQ + 20 Qufu^ 
( 30 ). 
From these again we find 
n^L^ = — 2ufLQ 
fl^Lg =: — ‘^ufLr.^ 
