DIFFERENTIAL INVARIANTS OF SPACE. 
293 
Jacobian set; as it involves 33 variable magnitndes, it possesses 15 independent 
integrals. Of these 15, nine are easily seen to be 
c", f, 9^ 
4*010’ 4*oov 
Other six are found to 1)e 
where 
a — 2L'(^ooo 
— 
Q (2/b(X) ''-^'oin) P (2Pioo ” ffijoi) 
b = 2 L‘^o2i, — P (2/^ 
010 
^fioo) Q^OIO “ ( 2/010 ^t)0l) 
c = 2L2 ^oo3 “ P {-0 
(101 
^loo) Q (2/(101 “ Aid) P'<^ooi 
1 
c> 
11 
JlOO + 9oio ffi ^9)01) ~ Q'^'ooi ” Pb)io 
S — '^^'4*101 Puffin 
— 
(AlOO “ 9^)\0 ffi Aid]) P'<^’lo0 
h = 2 L'<^i]q — P''b)io 
QAoO ■ P (./kkJ ffi I7(C0 " ^’OOl) 
P — 1 ^100’ 
/b 
9 = Af^inn + Hf^oio + 
' 4*0\0’ 
A 
f > 
1 ^Pim' 
/, 
c 
Q = ! 
9’ 
a, — + B/oin + F'/»(X)i’ 
r 
1 PolO’ 
f 
h 
ft 
. rool’ 
c, 
9 1 
P — ’ 4*100’ 
^b 
]i ' = Ct(/)]„o + P</>oiu ffi PAooi- 
4*010> 
h 
4*O0l) 
9’ 
f 
r 
( ’ 
14. A way of constructing these six solutions will l)e sufficiently illustrated by 
giving an outline of the process for any one of them, say the first. Tlie form of tlie 
ditferentlal equations suggests that tliey possess solutions, whicli are linear in the 
first derivatives of a, h, c,f[/, h and in tlie second derivatives of (p ; we therefore 
postulate a solution in the form 
^'^' 4*200 " “<^bfxj /3^100 ” yl/lOO “ VldO 
C^*0\0 %010 ^^010 ~ p/olo 
~ b/odi ““ xA)di 
where the coefficients (x., . . . , v do not involve first derivatives of o, h, c. f, <j, li or 
second derivatives of (p. In order that tliis expression may satisfy the three 
equations (II]), we must have 
acq^xj 
TO, 
010 
ea, 
laa. 
+ f/y “ 2L~f/) 
100 ’ 
2ha -b 1/3 + ./V = 2 L-(/>o] 0 ’ 
2r/« + + cy= 2U<P,,„ 
