767] 
ON THE GAUSSIAN THEORY OF SURFACES. 
335 
we have in all 10 determinants, viz. these are aa'a, = E'; aaa!, — F'; aa’a", = G'; 
aa'a"; and the six determinants aaa', aaa", act'a; aW, a'a'a", a'a'a. The foregoing 
expressions of E. 2 —F x and F.!—G x respectively, substituting therein for the determinants 
aaa, aa"a, aa'a", a'a"a their values as about to be obtained, are the required two 
equations. We have 
aa +bb +cc = E, 
a'a + b'b + c'c = F, 
aa + fib +7 c = \E X , 
a a -t- fi b + 7'c — %E 2 , 
a” a + fi"b + y"c =F, 2 -± 
aa + bb' + cc' = F, 
a'a' + b'b' + cc' = G, ■ 
aa' + fib' + yc' =F 1 —^E 2 , 
a'a' + fi'b' + y'c' =±G ly 
a"a+fi"b' + y"c' = ^G 2 ; 
and if from the first five equations, regarded as equations linear in (a, b, c), we 
eliminate these quantities, and from the second five equations, regarded as linear in 
(a, b', c), we eliminate these quantities, we obtain two sets each of five equations, 
a, 
a', 
a, 
a!, 
a" 
= 0, and 
a, 
a', 
a, 
t ft 
a, ol 
b, 
y, 
fi, 
p, 
fi" 
b, 
y, 
fi, 
fi\ 
fi" 
c, 
c, 
7> 
/ 
7 5 
<y" 
c, 
c', 
y, 
7> 
y" 
E, 
F, 
\E X , 
F a -iG x 
F, 
G, F x - 
\E„ 
\G X , 
\G 2 
These may be written, 
Fa a' a!' — \E x a!a a!' — \E,a!a!'a — (F. 2 — \Gfi aaa = 0, 
- Ea a! a!' + \E x a a! a!' + \E 2 aa!’a + (F. 2 - \G X ) aaa! = 0, 
Ea'a'a"- 
F a a! a!' + \E a G' - (F 2 - ±G X ) F' 
= 0, 
Ea'a"a - 
Faa"a - \E X G' + (F, — \G X ) E' 
= o, 
Ea'a a — 
F aa a! + \E X F' - %E,E' 
= 0; 
Ga a' a!' — (F x — \E,) a a a!' - G x a'a"a — ^ G 2 a'aa' = 0, 
— Fa a a" + (F x — ^E 2 ) a a a!' + \ G x aa"a + \ G 2 aaa' = 0. 
Fa'a'a" - G aaa" + ^G 1 G -\G 2 F' = 0, 
Fa! a"a - G aa'a — (F x — %E 2 )G’ + % G. 2 E' = 0, 
Fa a a - G a a a + (F x - %E S ) F' -\G X E = 0. 
Attending in each set only to the third, fourth, and fifth equations, and combining 
these in pairs, we obtain 
V 2 a a' a" + ( £FG X - FF 2 + lEG,) F' + (- \EG X + \FE 2 ) G' = 0, 
V 2 a’a a"+ ( i GG x - GF a + *FG X ) F' + (- \FG X + i GEJ G'= 0; 
V-a a"a + (- \FE X + EF l - ±EE 2 ) G' + (- \FG X + FF, - \EGfi E' = 0, 
V 2 a'a"a + (- \GE X + FF\ - \FE,fi G' + (- ±GG X + GF,_ - |FG a ) E' = 0; 
V 2 a a a! + ( \EG X - \FE a ) E' + ( \FE X - EF X + \EE 2 )F' = 0, 
V 2 a'a a' +( %FG X - | GE a ) E' + ( \GE X - FF, + \FE 2 ) F' = 0.