478] 
ON THE GEODESIC LINES ON AN ELLIPSOID. 
509 
Now (Leg. Fond. EUip., t. I., p. 71), we have 
II / (— k 2 sin 2 6) + II / ( — . 
sin 2 6 
F„ 
where, upon examination, it will appear that II / in fact represents the principal 
value of the integral. 
.... 1 k! 
Writing herein sin 2 0 = - >, and therefore cos 2 6 = - ., or tan 2 0 — k, this is 
1 — k 1 + K 
n, (— 1 + k) + II / (— 1 — k), = F /} 
and the formula (p'), p. 141, attributing therein to d the foregoing value, becomes 
(- 1 - O = E, + ~ |F / E(6) - E,F{ff)^ . 
But 6 is the value for the bisection of the function F /} viz., we have 
2 F ( 9) = F / , 
2E (6) = E / + l- V, 
whence 
F / E(d)-E / F(6) = i 2 (I- K ')F / , 
or the formula in question gives 
whence 
a ( _ i + k ') — 
the result which was to be proved. 
29. The value of M' (observing that 
which is 
b 
(a — b)(b — c) (Va — Vc) 2 « (1 - kJ 
dh 
is 
_ j\p = 4 
Va (1 — k) " ■' -a V/i (a + h) (c + h) ’ 
that is we have 
or, what is the same thing, 
that is 
log tan I (f)o 
tan \ <£ 0 
1 -k' 
~2~ 
1 — K 
(</>o the South azimuth of the 5-geodesic at the umbilicus). 
30. I purposely calculated the Table by quadratures as being a method available 
where the equation ac — b 2 = 0 is not satisfied; but in the present case, where this