DIFFEIJENTIAL lEVAKIAE^TS OF SPACE. 
285 
(^4^] 00 
dt 
[U^o]q 
dt 
^^<j>ooi 
dt 
^hoo^m + 4 >oioVioo ~r ^uoi^ioo» 
'^iOO^UlO “1“ ^OIO’^UIO ^OUl^iilO) 
'/’lUO^UOi ^01o'*7o01 “t" ^OOlC 01 ’ 
— 2'/).iqq^^qq + </>ioo^iOO '■^4’lO()V\QO ^‘iJlodiOO “1“ -''^liuClOO ^/'OOl^iOO’ 
dt 
djf)^ 
dt 
^^^11(1-2 
\lt ■ 
= 2f^iio4iu + ^lou^uju + -^o-ioVoio + 4>oiqdo-:o + -''/'oii^oio + ^-’oui^oiu’ 
= ^'^loi^uoi + ^lou^ouc + ^9^uu'^uui + ^l^oiodm + ^^/’uoiCoji + </>ooi4o2) 
'^-4’\ in 
'^^101 
(7/! 
((<f) 
01 1 
(/f 
= ^>110^^100 + ^Euu^uio + '/Eoo^iw + ^f^o-:oVm “t“ 4 >mdoio + ^oiu'^iiu 
+ ^/^oii^ioo “1" 4doi^oi(i '■!" ^/piuiCiiU’ 
= f/Eui^ioo + ^luu^ioi “i“ 4^onVm "1“ ^knolooi + ^uio^ioi 
+ ^oOi^lUO + ^lUl^ uui “t" ‘/5 )ui^iol> 
= ^loi^uiu + 4hi«^oo\ + ^Eoo^uii + 4^ouloio + ^h-2oVooi + 4>oioVon 
+ 9^o 02^()1U “1~ */^U11^001 ^/'uui-oiv 
The, Differential Idjuations characteristic oj the Invariance. 
7. Let o- denote any differential invariant involving at least some of the 
(juantities whose increments due to an infinitesimal variatio-n have just heen given. 
The differential equations characteristic of the invariance can he deduced from the 
e( Illation 
^ o- = nv' 
ill the usual manner : we suhstitute for each argument 6' in cr' its value 
e+fl>cu, 
and equate to zero the composite coefficient of dt on the nghtdiand side. The 
quantities t], i are arbitrary and independent; the coefficients of the various 
derivatives in the last equation must therefore vanish. these relations are ilie 
partial differential equations which, up to the order of differentiation retained, are 
characteristic of the invariants ; they are as follows : 
