804 
PROFESSOR A. R. FORSYTH ON THE 
Otlier invariants manifestly are given by 
{axt'a'y, ^{aVu){aVu'), ^ h{a"h"v)K 
and so on ; all of them are algebraically expressible in terms of the above aggregate 
of sixteen. 
1 he values of the indices of the respective invariants are given by the relation 
8 (/ -j- -j" 2 w?, -j“ n -|~ n' = 3g 
in each case. The results are as follows :— 
Index = 2 , 
1—1 
D20 
II 
-^135 
®125 
^■^13’ 
II 
A._,i, 
-^30 
00, 
©3, 
II 
OC 
^3, 
Ar3> 
^3- 
Tlie expressions of the 15 absolute invariants now are obvious. 
Invariants of the Third Order. 
22. The calculations, involved m this mode of constructing ditferential invariants, 
are very laliorioiis for differential invariants of the third order, being tlie order next 
in succession, d'hey are so extensive and demand such sustained attention merely 
through long processes of algebra tliat, if invariants of higher order are required, it 
will (in my opinion) he necessary to devise some other method of construction. 
Only the briefest outline of what has been done in the case of differential invariants 
of the third ordei‘ will be given, so far as conceiiis these laborious processes; the 
results will be stated. 
23. Tlie invariants in question involve derivatives of </> up to the third order 
inclusive; they also involve the quantities tq b, c, f, </, h, as well as their derivatives 
up to the second order inclusive. 
Ihe differential equations characteristic of the invariants are 57 in number; and 
they range themselves in three sets. The first set consists of the thirtv equations, 
which arise from the derivatives of t], { of the third order in the course of the 
process indicated in ^ 7 ; the second set consists of the eighteen equations which 
similarly arise from the derivatives of 77 , ^ of the second order; and the third set 
consists of the nine equations which similarly arise from the derivatives of 77 , ^ of 
the first order. 
The actual formation of the differential equations is effected as in § 7 . All that is 
needed for the purpose, in additnin to the results m 6 , is the aggregate of the 
expressions of the increments of the second derivatives of a, b, c, f, <j, h and of the 
third derivatives of ; and these are special in.stances of the formulte in § 5 . 
