186 
Proceedings of the Royal Society of Edinburgh. [Sess. 
is also covariantive ; and it remains covariantive when W and W' are the 
same function. 
30. On the basis of these two results, an algebraically complete system 
of concomitants of the required type can be constructed. 
Denoting tt(P) by u, we have a line-covariant v, where 
where 
_ 1 / du du du du du du \ 
^*^^3 wj 
= (D', M', . . ., A'(SP,, . . ., P,)^ 
D' = ND + SM + TL, 
E' = OE-f-UK-FPM, 
F' = VF-}-QL + RK, 
C' = VC + TH-f-UG, 
B' = OB + RG-i-SJ, 
A' = NA + PJ + QH, 
'2M' = NM -I- DP + SE + MO + TK + LU, 
2L' = NL-l-DQ-fSK + MR^-TF-^LV, 
2T' = NT -I- DH + SU + MG + TV + LC, 
2S' = NS + DJ + SO -f MB 4- TR LG, 
2K' OK -H ER -f UF + KV 4- MQ 4- PL , 
2U' = OU 4- EG 4- UV 4- KC 4- MH 4- PT, 
2P' = OP 4- E J 4- UQ + KH 4- M A 4- PN , 
2R' = OR 4- YR SQ 4- KB 4- GF 4- LJ , 
2Q' = VQ 4- NQ 4- RP 4- K J 4- HE 4- LA, 
2G' = VG 4- OG + TJ 4- HS 4- UB 4- RC, 
2H' = VH 4- NH UJ 4- GP 4- QC 4- TA, 
2 J ' = 0 J 4- N J 4- QG 4- RH 4- S A 4- PB , 
2N' = N2 4- AD 4- SP 4- M J 4- TQ + LH , 
20' = 02 4- BE 4- KG 4- UR 4- SP 4- M J , 
2V' V2 + CF 4- TQ + LH 4- KG 4- UR . 
Next, we liave a line-co variant tv, by forming 
dv ^ du dv du dv du dv du do du do' 
wd'WF^ sF,, 
= (D", M", . . AlP,, . . 
where, if we write 
we have 
P) TV ^ I A I ' ^ t ^ ^ 
D" = aD', M" = aM', 
for the whole sequence of coefficients. 
