168 
ON GEODESIC LINES, 
[508 
and it leads to a differential equation 
where 
P = (h + 2bp + fp 2 ) 2 — (a + 2hp + gp 2 ) (g + 2fp + cp 2 ), 
Q = (h + 2bq + fq 2 ) 2 - (a + 2h.q + gq 2 ) (g + 2fq + cq 2 ); 
and upon effecting the calculation, it is found that we have 
P = — 8oc 2 /3 2 (cf) 2 — 7 2 ) (a + 5 — 2c — (f>) (a+p) (5 +_p) (c +_p), 
Q = — 8a 2 /3 2 (c/> 2 — 7 2 ) (a + 5 — 2c — <j>) (a + q) (5 + 5) (c + q), 
viz. P, Q are the same multiples of (a+p) (b+p) (c + p), (a+q) (b + q)(c + q) respec 
tively ; so that, omitting the common factor, or taking P, Q to represent the last- 
mentioned functions respectively, we have 
dp dq _ 0 
VP VQ 
and since the parameter </> has disappeared, we see that the original equation involving 
(f> is the general integral of this differential equation; viz. that the differential 
equation belongs to the right lines on the surface. 
20. The form of the integral equation may be simplified by introducing instead 
of $ a new parameter K, connected with it by the equation 
T7 . fib — oca + (be . (/3 — a) K — fib + oca 
K = —zr 7—, or <6 = 77 — , 
/3 — 0C + (f> r c-K 
viz. we thence deduce 
¡3 — a + (f> — — 2ocfi 
(+). 
(-), 
(-), 
(-), 
(-), 
/35 — oca + (f>c — — 2 ocfiK 
(3b 2 — oca 2 — (p (a/3 — c 2 ) = 2a/8 (ac + 5c — a5 4- 2cP) 
</> + 7 = 2fi(K—b) 
(f> — y = — 2a (if — a) 
where the sign (-^) is used to signify that the functions preceding it have to be 
divided by a denominator which in fact is = c — K. The equation thus becomes 
(ac + bc — ab — 2cK — KX — Yf — 4 (K — a) (K — 5) (c 2 + cX + F) = 0; 
and if we moreover write 
v, p, \ = abc, be + ca+ ab, a+b + c, 
and instead of K introduce the parameter G, = X — K, the equation becomes 
{— p — 2c 2 + 2cG~(\—G)X— F} 2 + (5 + c — C)(c + a — C) (c 2 + cX + F) = 0;