178 
Proceedings of the Royal Society of Edinburgh. [Sess. 
U = (34, 13) 
S 
N = (14, 23) 
St ^ 
0 = (13, 42) 
V = (12, 34) 
f[14 
f][ 
■34 
,S 
13 
t 
24 
13" 
34 
ri2-| 
_ t _ 
s 
34 
s 
fr23i 
14 
24" 
13 
iL « J 
^ t _ 
s 
_ t 
r 
with the single relation 
n + o + y = o. 
These constitute an algebraically complete set of independent integrals, 
which simultaneously satisfy all the eighty equations of the first sub-group 
and all the forty equations of the second sub-group. (The quantity A is 
itself an invariant.) 
The Third Sub-Group : Sixteen Equations. 
24 . There remains the sub-group of sixteen equations which arise by 
equating the coefficients of the first derivatives of rj , 6 on the two 
sides of the relation 
</)' = {1 072-+- ^3-4^4) }</). 
The coefficients of Vi’ Vs ^ Vi ^ ^2’ ^1,^2’ ^3 all 
vanish ; three equations will be taken in the form 
CO. = CO. 772=^ CO. ^3 = CO. 6 ^, 
thus making a set of fifteen equations ; and a remaining equation 
CO. i^=€p(j) 
would serve to determine p. 
The equations in the set are linear and homogeneous, of the first order ; 
and they are a complete Jacobian set. The quantities that occur in them are 
(i) the fourteen variables Xj , . . .,X^; , . . .,04; P4 , . . ., Pgi 
(ii) the ten original coefficients a, h, . . n ; 
and, as the first and second derivatives can only occur in the combinations 
which so far have been called the Riemann-Christoffel symbols, 
(iii) the twenty-one quantities A, B, . . ., O, V, with a single 
relation between them. 
