176 
ON GEODESIC LINES, 
[508 
33. Next, for the skew hyperboloid, we have a and b = +, c = —, and I assume 
for convenience a > b. Attending to the signs, we still have therefore a >b >c. The 
principal sections are one of them an ellipse, and the other two hyperbolas, viz. the 
minor section is the ellipse — + y = 1, the major section is the hyperbola ~ ~ = 1, 
and the mean section is the hyperbola —— = 1: there are no umbilici. The elliptic 
cl c 
coordinates enter symmetrically; but, as before, we distinguish them, viz. we take p to 
extend from — c (a positive value) to infinity, and q from — b to —a. Thus p — const. 
denotes the curves of curvature of the one kind, viz. p = — c the ellipse — + y = 1, 
and every other value of p an oval curve surrounding the hyperboloid; and q = const. 
the curves of curvature of the other kind, viz. q = — c the major hyperbola j-—= 1, 
0 c 
q— — b the mean hyperbola —\— = 1, and each intermediate value gives a curve of 
CL C 
curvature of a hyperbolic form: we may say that p = const, determines the oval curves 
of curvature, and q = const, the hyperbolic curves of curvature. 
34. In the equation of the geodesic lines we have a + p, b + p, c + p, p all 
positive ; but a + q = +, b + q, c + q, q each = —; hence p h- (a+p) (b + p) (c +p) = +, but 
q -r- (a + q)(b + q) (c + q) — —; therefore 6+p and 6 + q must be of opposite signs, or 
we must have 6 + p = + and 6 + q = -; or what is the same thing, 0 may have any 
value from — p to — q, or say — 6 any value from p to q; that is, the value of — 6 
may be positive and greater than — c, positive and less than — c, negative and less 
than — b, negative and between — b and — a; viz. in the first case — 6 has a ^3-value, 
I 
o 
i 
B 
i 
Q A 
and in the fourth case it has a ^-value, but in the second and third cases it has 
neither a p- nor a g-value. This is better seen from the diagram. It follows that 
we have, on the hyperboloid, geodesic lines of four different kinds : those which touch 
a real curve of curvature, oval or hyperbolic, and those which touch no real curve 
of curvature, but for which — 6 has a positive value from 0 to — c, or a negative 
value from 0 to — b. And there are the transitional cases — 6 = — c, where the 
/£¡2 qj2 
geodesic touches the ellipse — + ~ = 1; 
0 = 0, where the geodesic becomes a right 
(X/ z* 
line ; and — 0 = — b, where the geodesic touches the mean hyperbola —1— = 1. 
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35. To explain this more in detail, consider the geodesics which start from a 
point M of the hyperboloid. To fix the ideas, consider the axis of z as vertical, and 
take the point M in the positive octant of the hyperboloid; and let Ml represent 
the direction of the oval curve of curvature, M9 that of the hyperbolic curve of 
curvature, Mb that of one of the right lines.