V 
[508 
B ( i + Gi q' 2 ); 
lit is 
ndependent 
or q, say 
ad q) only 
1 p and q. 
508] 
IN PARTICULAR THOSE OF A QUADRIC SURFACE. 
161 
7. Instead of starting, as above, from the equation given by Gauss, we may use 
the geometrical property that at each point of a geodesic line the osculating plane 
passes through the normal of the surface. 
Considering, as above, p, q as functions of a variable 0, then, 0 becoming 0 + d0> 
the new values of p, q are 
p +p'd0 + |p"d0 2 , q + q'd0 + \q"d0-; 
and that of x is 
x + a (p'd0 + \p"d0 2 ) + a' (q'd0 + \q"d0 2 ) + \ (ap 2 + 2a'p'q' + d'q' 2 )d0 2 ; 
or calling this x + x'd0 + ^x"d0 2 , we have 
x = ap' + a'q, x" = ap" + a'q" + op' 2 + 2 a'p'q' + a"q' 2 , 
and so 
y' = bp’ + b’q, y" = bp" + b'q" + (dp 2 + 2/3'p'q' + (3"q'% 
z' = cp + c'q', z" — cp" + c'q” + yp' 2 + 2ypq + y"q' 2 . 
The condition in question is expressed by the equation 
A, B, G =0, 
x', y', z' 
x", y", z" 
that is 
+ 
A, B, C 
ap' + a'q', bp' + b'q', cp' + c'q 
ap" + a'q", bp" + b'q", cp" + c'q" 
A, B, G 0. 
ap' + a'q, bp' + b'q', cp' + c'q 
ap' 2 + 2 a'p'q' + a"q' 2 , (dp 2 + 2 A'pq' + (d"q( 2 , yp' 2 + y 'p'q + y 'q' 2 
8. The first determinant is the sum of three terms such as A (be —b'c)(p'q" —p"q')\ 
viz. this is A 2 (p'q" —p"q), or the determinant is 
(A 2 + B 2 + C 2 ) (p'q" — p"q ), = (EG - F 2 ) (p'q" -p"q'). 
The second determinant is the sum of three terms such as 
(ap 2 + 2d p'q + a"q' 2 ) [B (cp' + c'q') — G (bp' + b'q')], 
where the factor in [ ] is 
p' [c (ca' — c'a) — b (ab' — a'b)] + q' \c' (ca! — c'a) — b (ab — a 5)], 
91 
C. VIII. L