188 Proceedings of the Poyal Society of Edinburgh. [Sess. 
Some of them can be obtained also in other fashions. Thus (ignoring 
a mere arithmetical factor) we have 
T — ?// 
' d 
a 
d 
d 
a 
-*-2 
ydp^’ 
dp,' 
dp,’ 
T — v( 
' d 
a 
d 
d 
a 
.dp^’ 
dp , ' 
' dp,’ 
^P2 
I - v( 
' d 
a 
d 
d 
a 
•■Slh ’ 
^Po' 
dp,’ 
dp,’ 
a 
d 
KPi ^ P2 rPs’ Pi^P5^ Pis) ^ 
^{Pi . P-2 PPz^Pa^ P5 ’ Pq) ’ 
and so on. 
The actual values of these invariants can be taken as follows:- 
b = N + 0 + V, 
which is permanently equal to zero, being the single relation among the 
coefficients of the fundamental line-covariant ; 
b = AD + EE H- OF + N2 + 0'^ + 
+ 2SP + 2JM + 2TQ + 2LH + 2GK + 2RU, 
which is the full value of Atq , viz. 
2N' + 20' + 2V' ; 
l3 = 2N" + 20"-|-2V" 
= N3-t-Q3 + V^ 
-f3N(AD+ MJ +SP +LH + TQ) 
4-30(BE-fGK -fPU+ MJ + SP) 
+ 3 V (CF 4- LH -1- TQ H- GK + RU) 
4- 3 A(MS H- LT) 4- 3B(MP -j- KU) 4- 3C(LQ 4- KR) 
+ 3D(HQ 4- JP) + 3E ( J S 4- GR) 4- 3 F(GU -|- HT) 
4-3J(KT 4-UL)4-3H(SK4-RM) 
4- 3Q(GM 4- US) 4- 3P(GL + RT) ; 
b - A'D' 4- B' E' 4- C'F' 4- N'^ 4- 0'^^ + 
4- 2S'P' 4- 2 J ' M ' 4- 2T'Q' 4- 2L' R' 4- 2GTv' 4- 2R'U' ; 
b = U"A'4-2M"J' + 2L"H'4-2rQ' 4-2S"P' +2N"N' 
4- E"B' 4- 2K"G' 4- 2U"R' 4- 20"0' 4- 2P"S' 
+ F"C' + 2U"U' + 2R"U'4-2Q"T' 
4- C"F'4-2G"K'4-2H"L' 
4- B"E'4-2J"M' 
4- A"D'; 
Ig = A"D" 4- P/E" 4- C"F" 4- N"- 4- 0"^ 4- 
4- 2S"P" 4- 2 J"M" 4- 2T"Q" 4- 2L"H" 4- 2G"K" + 2R"U". 
The full expression of b is long, that of Ig is very long. Instead of the 
latter, I prefer to take the discriminant of ^^(P) which differs from the 
