508] 
IN PARTICULAR THOSE OF A QUADRIC SURFACE. 
165 
where P', Q' stand for d p P and d q Q respectively; viz. this is 
p 3 . P 2 
+ p' 2 q' ■- PQ+(p-q) P'Q 
+ p'q' 2 .-PQ + (q-p) PQ' 
+ q' 3 . Q 2 
- 2 (p-q)PQ (p'q" -p"q') = 0. 
13. This has a first integral, 
P 
P 
0 +p 
+ 
Q 
6 + q 
= 0, 
where 6 is the constant of integration ; or say this is 
6 (p' 2 P - q 2 Q) + p 2 qP - q' 2 pQ = 0 ; 
viz. differentiating logarithmically, this gives 
2p'p"P — 2q'q"Q + p' 3 P' — q' 3 Q' _ 2p'p"qP — 2q'q"pQ + p' 2 (qp'P' + q'P) — q 2 (pq'Q' + p'Q) 
p' 2 P — q' 2 Q p' 2 qP — q 2 pQ 
which, multiplying out the denominators, is in fact the foregoing geodesic equation. 
To verify, consider first the part involving p", q": this is 
which is 
that is 
or say 
(2p'p"P - 2 qq"Q) (p' 2 qP - q' 2 pQ) - (2p'p"qP - 2 q'q'pQ) (p' 2 P - q' 2 Q), 
= 2p’p'P . q' 2 Q (q — p) — Zq'q'Q • p' 2 P (q ~p), 
= 2 (q — p) PQp'q (p' q ~ p'q"), 
= 2 (p-q) PQp'q' (p'q" ~ p"q')- 
We have next the part 
(p' 3 P' - q s Q') (p' 2 qP - q' 2 pQ) - [p 2 (qp'P' + q'P) + t WQ + P"Q)} (p'*? ~ q' 2 Q), 
which is readily found to be 
= -p'q' [p 3 P 2 + p' 2 q (-PQ+p- qP'Q) +p'q 2 (-PQ + q - pPQ') + q' 3 Q 2 } 
and the equation is thus verified. 
14. We have consequently 
d P\/( 
P 
(a +p) (b+p) (c +p) (0 + p) 
+ dq \/ 
(a + q)(b + q) (c + q)(6 + q) 
= 0, 
involving the arbitrary constant 6 as the differential equation of the first order of 
the geodesic lines on the quadric surface —f vl— = 1: the geodesic lines in question 
CL 0 G