172 
ON GEODESIC LINES 
[508 
We have, it is clear, (E x = d a E, E. 2 = d T E, &c.) 
4 4 
E, = ±-E, G, = --^-G. 
<7 + T <7 + T 
Hence the condition FG 2 — 2F 2 G + GGj = 0, in order that a = const, may be a geodesic, 
reduces itself to 
-~^~F-2F 2 +G 1 = 0; 
and similarly, the condition — 2EF l + E 1 F+EE. 2 = Q, in order that t = const, may be a 
geodesic, reduces itself to 
Ji— F— 2F, + E 2 = 0. 
<7 + T 
We have at once 
Е 2 = 4<а(т 2 — 1)(тсг + 1) — 46 (т 2 + 1) (та — 1) +4с.2т(сг— т) (ч-), 
= 4а (сг 2 — 1) (т<7 + 1) — 46 (о- 2 4- 1)(т<т— 1) — 4с.2сг(сг— т) (-=-), 
F 1 = 2а (т 2 — 1) (тег — а 2 + 2) — 2Ъ (т 2 + 1) (тег — сг 2 — 2) — 2с. 2т (т — Зет) (ч-), 
F 2 = 2а (о- 2 — 1) (то- — т 2 + 2) — 26 (о- 2 + 1) (то- — т 2 — 2) — 2с. 2о- (о- — Зт) (-г-), 
where denom. = (о- + т) 5 ; and substituting these values, the conditions are verified: we 
thus again see d ■posteriori that the right lines a = const, and т = const, are geodesics. 
25. The last-mentioned values of E, G are E = T + (t + o-) 4 , G = "Z -f- (t + o-) 4 ; and 
writing for a moment 
A = a (т 2 — 1) (a 2 — 1) — 6 (t 2 + 1) (o- 2 + 1) — с. 4o-t, 
we have F= A-=r (t + o-) 4 , the value of ds 2 is thus 
= Tdo- 2 + 2 Ada dr + Idr 2 (t 4- a) 4 , 
which should be 
where, as before, 
= 4p- 
dv 2 da 2 
P-p-i Q 
P, Q = (a+p) (b+p)(c +p), (a + ?)(6 + 7)(c + 2), 
respectively. We have already found 
dp dq Ыа 
or, what is the same thing, 
VP + VQ + V2 °’ 
dp dq 4 dr _ 
vr - ve 7r~ U; 
vp 
dp _ g / do - dr \ 
__ wx + vt; 
do- dT 
vs + vt;’ 
a? __ 9 
V<2~ 
V2 
dT\