IN PARTICULAR THOSE OF A QUADRIC SURFACE. 
175 
[508 
508] 
divided by 
ies on the 
,r lines. 
right lines 
the sides 
sfficients of 
is possible 
t being at 
vvhich have 
atio be as 
; ± m VS dr, 
,’allelograms 
curvature, 
audace into 
/e the two 
s theorems 
as to the 
-mentioned 
•ately. 
assume as 
}he major- 
mean, or say the minor section, 
y- z* 
w4 - 
b c 
= 1 the minor-mean, or say the major section, 
and —|— = 1 the mean, or umbilicar section. The elliptic coordinates p, q enter into 
CL G 
the equations symmetrically, but we distinguish them by taking p to extend from — c 
to — b, and q to extend from — b to —a. Thus p = const, denotes the curves of 
c 
curvature of the one kind; viz. p= — c denotes the major-mean section ~ = 1, 
p — — b the portions UTJ' and U"U'" of the umbilicar section; and q — const, denotes 
the curves of curvature of the other kind, viz. q= — b the remaining portions U' U" 
and U'"U of the umbilicar section, q = — a the minor-mean section ^ 4 “ = 1; say 
p = const, the major-mean curves, and q = const, the minor-mean curves. 
32. Hence, in order that the equation 
^ \J((a 4 p) (b 4 p)(c 4 p) (0 4p)) ~ ^ \j((« 4- q)(b + q)(c + q){0 + <?)) 
of the geodesic lines may be real {observing that we have a+p, b+p — +, c + p, 
p = ~, and a + q=+, b + q, c + q, q = ~, consequently p a (a +p) (b + p) (c + p) = +, but 
q + ( a + q)(J ) + q')( c + q) = -} ) W e must have 0 + p, 0+q of opposite signs, that is 
0 +p = + and 0 4 q = -; or 0 included between the limits a, c. Or, what is the 
same thing, — 0 is included between the limits — c, — b, say — 0 has a p-value; or 
else between the limits - b, -a, say - 0 has a q-value. This is conveniently shown 
0 G P B Q A 
in the annexed diagram of the values of —p, —q, —0• Hence on the ellipsoid we 
have two kinds of geodesic lines, each of them touching a real curve of curvature, 
viz. those which touch a major-mean curve and those which touch a minor-mean 
curve: the transition case, answering to the value 0 = b, is that of the geodesic lines 
which pass through an umbilicus. I have considered the theory more in detail in my 
memoir “ On the Geodesic Lines of an Ellipsoid, Mem. R. Ast. Soc., t. xxx., pp. 31 53, 
1872, [478].