OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
S81 
Let 
1), = (E (G^ - 2Fy + ES) - F (G« - 2F;8 + Ey)' + . . . ; 
then as 
E^ (a'^d — Sa/3y + 2^’^) — Ea~ ■) E~S — .SEFy + (EG -|- 2B'') j8 — FGa [■ 
= — 3Ea (E/3 — Fa) (Ga — 2F/? -F Ey) 4" - — Fa) ”’ -F 2 VG- (E,8 — Fa), 
we liave 
(a’4 — (an, iv^^) — 2J'^ (/'n, ay) — 2VhG'J (ayvay) = 0. 
Consequently G is expressil)le in terms of members of the S 3 ^stem and of (ay) ; and 
<1> [w^ is expressilde in terms of a’.^, H (ay), I (rt^j.), J (ay). When the various 
expressions are sidostituted, we can modify the system of concomitants in § 21 ; v’e 
replace an}^ one of tliem, say 1 (ay), by h,, and tbe modified aggregate is still 
complete. The expression for (ylj then gives the significance of ly,, tlie index of 
which is 5 ; wheji the values of tlie invariants already interpreted are substituted, 
we find 
_ T5’ i _ K , t dBdHl 
W Idad.s’ F"^Bds’(/ar 
Similarlv, we liave 
v-^ 
FI (a/ 
Y+ 
or., 
and thus we obtain another expressifar for F,V ^ in the form 
F, _ r (/m _ K 1 dHl 
[(ls<h r' p" ils ]■ 
Comparlim tlie two values tlius obtained for h„ we at once have an exiiression for 
dm dm . , 
07 .fc - '’y 
(FH _ (FH ^ _ 1 1 FB FH 
dsdn duds p" Fi' B ds die 
yn 
introduce a covariant fy of index 6, defined as 
h,= |FF(Gy-2Fd+Ee)-2EF(G^-2Fy+E8) + FMGa~2F/3+Ey)( .^oF+ • • • ; 
this covariant is expressible in terms of ay, H (ay), I (ay), J ('J'^), J (a.y, ay) and, as 
To simplify tbe system we 
44. We proceed in tlie same way witli 
Fa, 
