160 
ON GEODESIC LINES, 
[508 
5. We have 
M = (p'dE + qdF) n - (Ep' + Fq') (%p' 2 dE + p'q'dF+ \q' 2 dG), 
= dE {p'il — \p 2 (Ep + Fq)} 
+ dF [q’Gl - p'q' (Ep' + Fq')} 
+ dG{ -^(Ep' + Fq% 
or say 
2 M = dE [ p' 2 (p'E + q'F) + 2p'q' (p'F + q'G)] 
4- dF [ 2q' 2 (p'F + q'G)] 
+ dG[-q' 2 (p'E + q'F) ], 
= (E x p' + E, q') [ p' 2 (Ep' + Fq') + 2p'q (Fp + Gq')] 
+ (F lP ' + F 2 q')[ 2q' 2 (Fp' + Gq')] 
+ (G lP '+G 2 q)[-q'- 2 (Ep' + Fq') ]. 
The term in p' 4 is EE^E, which is also the term in p' 4 of fl (E lP ' 2 + 2F 1 p'q' + G^/ 2 ); 
whence 11 (E 1 p' 2 + 2F 1 p'q + G x q 2 ) — 2M divides by q. 
Proceeding to the reduction: 
Term in E x is 
E-y .p' 2 11 — p' 3 (Ep' + Fq') — 2p' 2 q' (Fp' + Gq), =E 1 .— p 2 q' (Fp + Gq); 
term in Fy is 
Fy. 2p'q'n - 2p'q' 2 (Fp' + Gq), = Fy. 2p' 2 q' (Ep' + Fq') ; 
term in Gy is 
Gy. q'H2 + p'q 2 (Ep' + Fq'), = Gy [2p'q' 2 (Ep' + Fq') + r/ 3 (Fp' + Gq')}. 
6. The remaining terms in E 2 , F 2 , G 2 require no reduction, and the result is 
Ey {-p' 2 (Fp' + Gq)} - E 2 { p' 2 (Ep' + Fq') + 2p'q' (Fp' + Gq')} 
+ Fy {2 p' 2 (Ep' + Fq')} - F 2 {2q' 2 (Fp' + Gq')} 
+ Gy {2p'q' (Ep' + Fq') + q' 2 (Fp' + Gq')} - G 2 {- q' 2 (Ep' + Fq')} 
+ 2 (EG- F 2 ) (p'q" — p"q') = 0, 
or, what is the same thing, 
(Ep' + Fq') {(2Fy - E 2 )p' 2 + 2Gyp'q' + G 2 q' 2 } 
- (Fp + Gq') [Eyp 2 + 2E 2 p'q' + (2F 2 - Gy) q' 2 } 
+ 2 (EG- F 2 ) (p'q" — p"q) = 0, 
which is the required differential equation of the second order: the independent 
variable has been taken to be the arbitrary quantity 6; but taking 6—p, or q, say 
6=p, we have p = 1, p" = 0, and the equation then contains (besides p and q) only 
q" and q', that is, d p 2 q and d p q, and is therefore a differential relation between p and q.