HYPERBOLOID OF REVOLUTION. 
239 
592] 
As already mentioned, u is connected with the parametric latitude -»Jr by the 
equation 
. c u 
sm ÿ = - 
that is, 
or conversely 
_u>J(e 2 — 1) 
a*J(l-u 2 )’ ~ V(1 - u-j ’ 
sin , \J/' = \f(ß 2 — 1) tan p, if u — sin p, 
sin i/r 
u = 
V(e 2 — 1 + sin 2 ’ 
so that the point passing to infinity along the branch of the hyperbola, or y}r passing 
from 0 to 90°, u passes from 0 to -; and for u = - the value of Z becomes, as it 
should do, infinite. The value of z in terms of u is 
(e 2 — 1) u 
z = 
V(1 - e 2 u 2 ) 
, or conversely u = 
sj{c 2 z 2 + e 2 — 1) ’ 
and we have, moreover, 
u = 
_ 1 2Ç 
sin-vjr 
^, = -sin <f>, = (as before) -tt——-— 
1 + £ 2 ’ e r v ’ \/(e 2 -1 + si 
sin 2 Ÿ) ‘ 
It will be recollected that, in the Mercator’s-projection of the sphere, the longitude and 
latitude being 0, </>, the values of X, Z are 
X = ad, Z— log tan ^ + i</>j , 
the logarithm being hyperbolic. 
In the case of the rectangular hyperbola a — c, =1 suppose, 
e = V(2), z = tan i/r, u — 
sin yjr 
\/(l + sin 2 -^) 
, = sinp, if sin yjr = tanp ; 
whence 
/7 it 7 tan (45°-^) , /(e)) k z tan (22°30 / - jp) 
• 1 tan (45° + ip) " 2 } ’ tan (22°30' + ip) ’ 
the first term being of course 
= h.ltan (45° — ip), or -h.l tan(45° p ip). 
Transforming to ordinary logarithms, this is 
V(2) log e 
say this is 
[- V(2) log tan (45° + ip) + {log tan (22°30' + ip) - log tan (22°30' - ip)}]. 
V(2) log e 
A + B),