DIFFEKKNTIAL INVARIANTS OF SPACE. ^ 
;I17 
«io - 
/^JO — 
7io = 
^10 — 
6L- ~ + 1'' (^/iioi Olio) 
C)Tr {2/7,1,~ (7,1,^) + B (2/,|,i| — Oil,,) + i" Oioi}’ 
GL' (A (2/7,-,Qi — ej,,n) + H (2Aii)i — t'oio) + 
2B (2(?qiqA,„i]^ — t'oio/ioo + Aiol/oio " \tooJhyn H“ -/m/wi ~ -.Aioi.Viiin) 
4 - 2 A ( 2 aQo;^f’j(jn ■^'■-^ooi.Vuoi) H“ (/’inrAini ” '^'-ooi.Aioi) 
+ 2F (— 2c,)(,j^/(„oi + Aioi/ioo “ Oioil/oio+2tNoo/ini ■“'TniAni) + 'V/i,,ii(',)i(| —<l/i«ii.Vn()i) 
A 2 Ct (— 2«,,|-);^(:'qq]^ a ^^'io<'9ooi " '' 100 “ *"’1/ ooi) 
+ 2H {2r,J> 
1)01 
I’lDo,/^ 100 "B * loii.Viiin “B oio 
— '^'"'ooi/ooi "" B 7 i,i,i//|| 0 i “B 24loR/uiii 
“■//ol(I/ii(ii)> 
/^lo = 
^^10 = 
2A (2t’iij|,AQO[ — I’loo.Voio "B <^’joo/ioo B/ooi^'ooi "B ^//oiol/ooi 24i,„,//o()i) 
-{- 2B (2/i,j,,^c‘||]^q '^:^A)0KAini) "B 2C (I'oirVooi ’^'^ooi.Aioi) 
+ 2F(~ 26 |jQ^r,|,„ + — c7;)|q F^'ooi) 
+ 2 G (— 2 ('|jq,^//|),)^ + Aoil/uio “ Cjoi/100 "B’ 20 )ioi/ooi 'TooOiio "Blooyooi Q/om.Vooi) 
2 H ( 2 ('|||QA,|f,i i^'dio.Voio "B 0)10./100 “B -'^OoAioo 
-^^Aioi.I/ooi —’ A/ooi/'ooi "B 2/001I/010 “ 2 /i(io 4 iiui)> 
2A(A\co “ '"loof/uoi ” ^I/'ooi) + 2B(7'7 ,i,) — 0);o,/ooi “ ^./Oioi) ~ 
“B 2i (/’oloOjOl A 2Ct ((’inoOxii "*0)01.7001) 
A 2H (2rj,-,f,r’„2f, Ojlo.7u01 ''lOO./oOl B/onlf/onl)' 
2-1. Before proceeding to the third set of eipiations, one sidistantial simplification is 
possible. The various quantities that have been obtained are algebraically independent, 
so far as they occur as solutions of partial difterential equations. But there may be 
intrinsic relations among them owing to the original properties of tlie quantities which 
they involye; such intrinsic relations are known to exist among the diHerential 
invariants of a surface. 
As a matter of fact, each of (he .six cepialwns 0,1) B3, 0 |, Gji ©r, is epial to ze)’0.i 
a result e.stablished'^' by (/ayley in a somewhat different form. The six equations 
that thus arise, in the case where the independent vanahles u, r, w correspond to a 
triply orthogonal system, are the well known six relations givenf liy Lame. 
As the six quantities 0^, 0,>, 0o, 0^, 0^, 0^ are permanently zero, we may use this 
property to simplify the equations in tlie next set. 
The evanescence of these quantities might have been used earlier, in order to 
modify some of the preceding expressions ; but no substantial advamtage Avould have 
* ‘Coll. Math. Paper.^,’ vol. xiL, pp. 12, 1-3. 
t A reference is made by C.vylky, Jor. cit., p. 17. 
