DIFFERENTIAL INVAPJANTS OF SPACE. 
323 
The expression of Ig is obtained by substituting 'i?, (5, for A, B, C, F, G, H 
in Ig throughout ; and the expiession for is obtained similarly by substituting 
A, B, C, F, G, H for A, B, C, F, G, H in Ig throughout.* 
28. All the 17 differential invariants so far given are only relative invariants; it 
remains to obtain the index of each of them, so as to obtain the absolute invariants. 
These indices are given by th,e formula 
op, — j) 2q -I- 8r -j- 16*' 
of § 25, and are found to be as follows :— 
Index 
.) 
Ini 
I3, I 
h, 
4 I, 
8, 
Iri 
Is; 
il, 
ly; 
IG, 
Iim 1 
18, 
Ill, ; 
20, 
I12; 
22, 
I17 ; 
2G, 
I15; 
30, 
Ii,r 
in 
4.3 ’ 
Accordingly tho'c are .•ii.rteeii alyebraically independent ahsoluic differentied 
invariants vp to the specified order in the derivatives of a, h, c, f, (j, h and the 
derivatives of </>; and an algeliraicahy complete aggregate of these invariants is 
provided by the set 
I I -1. ] I -;j 11--- I r r 1 • r r • i t • r i • 
1 I -s T T -8 I T -8 . I r -y . I -lu . I r -ii . i t -13 . r r -i5 
(Jeornet)'ic Signidicanee of the simpler Invariants. 
29. \¥e proceed now to obtain the geometric significance of some of the differential 
invariants which have been obtained. 
The only in variant which is free from derivatives of a, b, c, f y, h, (f), <f>', f is the 
ipiantity, denoted sometimes by L'^ and sometimes by A^. We have 
Aj^ = = aba -j- '2fyh — af^ — by^ — eld ; 
* A factor 2 needs to be dropped from B and ho in passing from the symbolical form to the developed 
form. 
2x2 
