OF A SUKFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
375 
and therefore 
= (GP - 2PQ + ER)^„, + (GQ - 2FR + ES)(- <!>,„) 
= «l> 
say, where is a covariant of the system with index easily seen to be equal to 3. 
Now it is easy to verify that 
(EQ - FP)^ = (GP - 2FQ + ER) EP - (EG - F^) P^ _ (PR _ Q3) e^, 
and therefore that 
W 3 ) = w.2iv.;^a^ — — ?ro~H {ni^. 
Consequently is expressible in terms of the members of the system ; when the 
expression is substituted al)ove, the result enables us to ol)tain the value of 
H (/Cg) But it is simpler to modify the original system of concomitants in § 21 : 
we can replace H (^Cg) in that aggregate by a]_, and the modified aggregate still is 
complete. For the significance of iq, we have 
til _ 
v» = ® * • 
Further, we have 
yvw = (GP - 2FQ + EPv) (- F,^„, + 04 ,,,,} 
fi“ (^^Q — 2PR -|- ES) (E(^,^i — 
— ^2j 
say, where is a covariant of the system with index easily seen to be 4. It is easy 
to verify that 
E3 (P2S - 3PQR 4- 2Q3) - EP3 ^E^S - 3EFR + (EG + 2F2) Q - FGP'- 
= - 3EP (EQ - FP) (GP ~ 2EQ + ER) + 2 (EQ - FP)^ + (EQ - FP), 
and therefore 
^^^(^(iCg) — w.2W^\ + {w^, w^) — w^) — 2V2«q2J (wg, w^) = 0. 
Consequently tq is expressible in terms of members of the system; when the 
expression is substituted above, the result enables us to obtain the value of <F (nq) V'®. 
But, as in the last case, it is simpler to modify the original system of concomitants in 
§ 21 : we can replace ("?%) m that aggregate by aj, and the modified aggregate still 
is complete. For the significance of a 2 , we have 
^3 _ E 
V" dn ' 
