ON GEODESIC LINES, 
162 
[508 
which is —p' (a'E — ciF) + q' (a'F — aG). We have thus the second determinant, and 
the equation becomes 
(■EG-F 2 )(p'q"-p"q') 
+ (op’ 2 + 2a'p'q' + a"q' 2 ) [p' (a'E — aF) + q (a'F — aG)) 
+ (/3p' 2 + 2/S>Y + P"q' 2 ) {p' (b'E - bF) + q' (b'F - bG)} 
+ (yp'* + 2yp'q' + y"q'*) {p (c'E - cF) 4- q' (c'F — cG)} = 0, 
an equation of the same form as that previously obtained, and which can be without 
difficulty identified therewith. 
The Circular Curves are Geodesics. 
9. I proceed to show that the circular curves are geodesics; viz. that an integral 
of the geodesic equation is 
{E, F, G\p', q') 2 = 0. 
Starting from this equation, we have 
2 {(Ep' + Fq')p" + (Fp + Gq') g"} + (E x p' + E 2 q')p' 2 + 2 (F x p + F.q)p'q' + (G x p' + G 2 q') q' 2 = 0. 
Now the equation, writing therein Ep' + Fq = \q, gives Fp f Gq = — \p': these equations 
may be written 
Ep' + (F-\)q' = 0, 
(F + \) p' + G q = 0, 
the value of A being therefore -A 2 = EG — F 2 . The result just obtained thus becomes 
2X (p"q — p'q") 
+ [(Epi + Ej() p + (Ftp 1 + F.// ) q ] . - I (Fp + Gq') 
+ PVp' + Fjftp' + (<?y + G4) ?']. i (Ep' + Fq). 
that is 
2 (EG — F 2 ) (pq" —p"q') 
- (Fp' + Gq') [E lP ' 2 + (E % + F l )p'q' + F 2 q' 2 ] 
+ (Ep + Fq') [F x p 2 + (F 2 + Gi)p'q + G»q' 2 ] = 0 ; 
or what is the same thing, adding hereto the zero value 
A (E, F, G%p', qj, = (Fp' + Gq')Aq' + (Ep' + Fq')Ap', 
where A is arbitrary, the equation is 
2 (EG — F 2 ) (p'q" —p"q') 
+ (Ep + Fq) [Ftp' 2 + (F 2 + Gq)pcq + G 2 q' 2 + Ap] 
- (Fp' + Gq') [.E x p' 2 + (Eq + Fq) p'q' + F.// 2 - Ar/] = 0; 
viz. taking A = (Fq-E.2)p'+ (Gq—F,) q', this agrees with the geodesic equation. 
The foregoing integral, (#, F, GQp', q')-= 0, is, I believe, a particular, not a singular, 
solution of the geodesic equation.