o22 T’IJ0FP::^S0I; a. 11 . FOK.SYTH DX TIIF 
27 . All these exin'essions are ^iveu In iimbral symbols. Considerable labour mio-bt 
be involved in the j^roce.ss of transforming them so that the changed expre.ssions 
involve the real symbols; but it is possible to avoid most of this labour bv remem¬ 
bering that all the functions are invariants and contravariants of a sv.stem of ternaiu' 
*'■ «. 
forms, many of which have been calculated and are tabulated in connection with the 
theory of homogeneous forms. Accordingly, the outline scheme of the real expressions 
is as follows. 
The explicit non-umbral expressions for the first seven of these differential 
invariants have been given in § Ih. Tlie invariant is given in explicit form by 
Cayley, who denotes"^' it by PU. Tlie invariant Ij- is given in explicit form by 
Cayley, who denotest it by QU. The invariant Ij- is given in explicit form by 
Cayley, who denotes! it by FU. The invariants Ijn and are given in explicit 
form by Cayley, who denotes^ them by S and T respectively. The invariants Ij3 
and I34 are obtained by operating on with the two operators given at the end of 
§ 26 . The actual expression for Ig is obtained by developing the uinbral form : it is 
found to be 
Ig = {A (bV' - f"-^) + B [W - r^) + C{V'n" ~ m"^) 
+ A(kV' - IV) + G (bV -f cT' - 2fV) + + 
+ {A (aV' - + B{\iV - f"~) + (7(g"n" - c"-) 
-h F{W + gV - 2fV') + Cf (aV'-cYO + ^^(c'V + a^'m''- 2fY')]?C' 
+ {A (a'V - h"-) + i?(b"k" - b"~) + e(gT' - f"-) 
+ A(hT' -f g'V - 2b"f") -f G (a'r + b'V' - V) + H (A'k" - bV) j 
+ 2 ) A (g'V - a^r) + B{yT - hT') + C {c'T - gV) 
+ F{r- + bV' - ff'in" - gT') + G {c"F' ~ a"m") -f T/(b"g" - 2 .’V)]iuu^ 
+ 2 [A (fli" - b"g") + B{yr - n”) + C{m'T - c'T) 
+ A(b"m" - c"k") + G{r- + b"m" - V'g" - b"c") + 7A(r'h" - g"k")l?/ 3 t^i 
+ 2 {A (fV - cV) + if (fT'- ff'm") + e(c"m" - f"n") 
+ F{c'r - bV) + G (g"m" - b"n") + IJ {f'- + V'g'' - ff'm" - V'c")]u,u,. 
* ‘Coll. Math. Papers,’ vol. 2, p. 326. It should be mentioned that the quantities a, h, c, f, g, h, i, j, l\ 1 
in Cayley’s memoir are to he replaced hy a", k", n", 1”, c", h", ui", g”, h ", f” respectively, that the 
quantities g, ( in Cayley’s memoir are to he rejilaccd liy ?q, ?q>, ?/-„ ami that bi contains a numerical 
factor 6 which can be rejected. 
t ‘ Coll. Math. Papers,’ vol. 2, p. 327, vrith similar modifications and rejection of a numerical factor. 
J ‘ Coll. Math. Papers,’ vol. 2, p. 328. 
§ ‘ Coll. Math. Papers,’ vol. 2, p. 32-5. 
