DIFFERENTIAL INVARIANTS OF SPACE. 
8(:)i 
Of these 4 , two are already given by and ©p Let 
c 
2 D = (f)\cjO -^T- 1 " ^'oio AT- 1 " ^'ooi 
_a_ 
^^OUl 
^4>ino ' ^ ^<^010 
then other two are given by D©j and D"©^ The four quantities 
A|, ©1. D©^, D'©j 
are algebraically independent of one another ; and so they constitute the recphred 
aggregate of relative differential invariants. 
Similarly, we can obtain an algebraically independent aggregate of differential 
invariants, which involve a, h, c, /, g, h but none of their derivatives, and winch 
involve first derivatives of three quantities (f), </>', (f)" unconnected by any identical 
functional relation. Let 
2D' 3 . . 
c 
+ --.I + ^ ooi 
ino 
)i(i 
101 
then we have, as simultaneous invariants, 
Ai, ©„ D©i, D%i, D'©i, DD'©i, 
and 
not difficnlt to prove that 
T = 
^100’ 
4 * 100) ^ 
^Aoiih 
um> 4 
^01 in 
4 * non ^ 
L invariants in 
the algi 
<'“>n 
D©j, 
D'©i 
D©j, 
I»)„ 
DD'©, 
D'©„ 
DI)'©i, 
mil 
(im 
1)111 
= At-P, 
so that the above eight (piantities, subject to this one relation, constitute tlie 
aggregate. 
20. We proceed similarly with tlie determination of the invariants of higher orders 
associated Avitli tv o surfaces : here, we shall restrict ourselves to the consideration 
of those invariants which involve derivatives of a, b, c,/, g, U of the first order and 
also involve derivatives of two (piantities j) and of order not higher than tlie 
second, 'fo obtain the expressions of suvli differential invariants, we take six new 
quantities 
a' = 2L"'V^'2fjQ — P cqiiii Q ftim) ( 2 (/io(i ^tioi) 
b' = 2 L^^'q2o — P'(2/qijf, ^loo) Q D (2b,,j,, ^Aio]) 
c' = 2 L''i^',iq 2 — P (2pi|Qj Ciiifi) Q ( 2 _/(i(ii '^'om) P ^1101 
P = 2 L^^'qii — P^ ( —^100 + .'/oio + ~ Q^^ooi ~ fil^Cim 
— 2L~(^';[oi P^^^ooi Q (./loo 5^010 "P ^%ii) *^100 
P^^oio Q ^100 P (yioo "k 9 o\o ^%ji) 
L 
( ’ 
