rROFEssoi; a. r. forsytii ux the 
•292 
1]. Proceeding as in § 9, we can similarly prove that there is no proper differential 
invariant of the first order in the quantities a, h, c,f, p, //, which involves the first 
deiivatives of (f), that is, that L" and © are the only invariants which involve the first 
derivatives of (f), the quantities a, h, c, f, g, h, and their derivatives of the first order. 
lnvavi<infs of the Second Oixlrr. 
12. We now proceed to consider the aggregate of algebraically independent 
differential invariants, which involve a, b, c, f g, h and their first derivatives, and 
which also involve derivatives of (f) iq) to the second order inclusive. It has been 
proved that when no derivative of <f) occurs, tliere is only a single such invariant, viz., 
Id; and that there is one such invariant involving the first derivatives of (f), viz., ©. 
ffhe equations characteristic to the invariants are the set given in § 7 ; their number 
is 18 + 6 + 3 = 27 . Taking the last three in the ecpiivalent form 
A-(o-) — Ag (o-) = 0, 
A;(cr) — — 0, 
A^ (cr) + Ag {( t ) -f- Ag (ct) = 3p(T, 
and associating the first two of these with the earlier 18 -p have 2G equations 
in all. They are linearly indeptendent of one another; and they form a complete 
Jacobian system. Tlie nundjer of variables that occur is 
G, for the Cjuantities a, h, c, f g, h, 
+ 18, for their first derivatives, 
+ 3, for the first derivatives of (f), 
+ G, . . second . . . (p, 
l)eing 33 in all. There are therefore 7 solutions common to the 2 G equations ; two of 
tliem are already known, being L" and 0; and therefore otlier five are required. It 
is manifest, from the manner in which these five have been selected, that each of 
tliem must Involve second derivatives of cf). 
Tlie remaining equation will be found to be satisfied for each of the five solutions 
by the appropriate determination of the index jx in each case : the actual determination 
is made in a simple manner, owing to ]n’operties of liomogeneity possessed hy the 
solution. 
13. The mode of manipulating the eipiatloiis, so as to obtain an algebraically 
complete aggregate of integrals, is similar to the mode in the corresponding investiga¬ 
tion of the differential invariants of surfaces. 
We begin by obtaining the proper number of Independent integrals which belong to 
tlie set of eighteen equations (IT,), . . . , (II,-,). This set is, in itself, a compilete 
