DIFFERENTIAL INVARIANTS OF SPACE. 
305 
The first set of equations, consisting of 30 members, possesses sixteen independent 
integrals, as well as a, h, c, /, g, h, their derivatives of the first order, and the 
derivatives of 6 of the second order. The sixteen integrals are :— 
^''002 
+ 
^020 
uin 
^200 
+ 
^'002 
^02U 
+ 
^200 
'^011 
— 
^110 
'^hoi 
fi~ y2oo> 
^101 
— 
^hji] 
.Aio 
”R .%20’ 
e,= 
^' 1 LO 
— 
/lOl 
9o\\ 
+ ^5)02 ; 
^q = 
2L2 
4*m) ~ 
I ^2UU 
— Q 
'-^iio) “ {'^9200 ^hoi)> 
u.j 2 L (f>2Hj 
Uo = — Q ( 2 /qi^ — — Rc^qu, 
P'^Otio Q^iio R — *^101)’ 
~ Q^iui ■“ 
^^ 4^10-2 ^ 'An )2 Q (^/lul ~ ^’ 110 ) R^'i()i5 
^/*080 R Q4^02I) (^/o 20 ” *^0!l)> 
u, 
«G 
u~ 
i 
Uc 
Ua 
2 L R(^^5ni Q^un — 
U 20 J 
4*012 T {^9oil ^‘ 110 ) Q^U 02 ~ POjll; 
'^hu R(‘-^1 /()i)2 ^'liil) ~ Q (‘-^/uoS “ 0 )1 i) ~ R0j()2 - 
All these quantities are integrals ol tlie 30 equations, as also are all functional 
combinations of them. Ihe integrals of the remaining equations must accordingly be 
some functional combinations of 61 ,.. , 6 ^, u^,. . , as well as of a, h, c, f, g, h, their 
deiivati\ es of the first order, and the derivatives of of the second order. 
The second set of equations, consisting of 18 members, possesses sixteen independent 
integrals, in addition to a, b, c, f, g, h, a, b, c, f, g, h, and tlie derivatives of ^ of the 
first order. The sixteen integrals are :— 
VOL. ocii.— A. 2 n 
