5 
The abscissa is then expressed by 
X = Uja + u' 
and the ordinate by 
y = Z sin Q£ 
y = r cot u sin a 
or 
(4) 
(5) 
When the projected points are situated to the north of the axis of 
abscissae, that is to say, when arc v is smaller than 30 degrees, the colatitude 
of the origin of co-ordinates, the y values are given the sign ; when the 
projected points are to the south of the axis of abscissae where arc v is 
greater than 30 degrees, the ordinates y are distinguished by the sign — . 
The ordinates are to be plotted accordingly above or below the axis of 
abscissae. The x values are scaled east and west of the central meridian. 
Each parallel of latitude is thus separately computed, developed, and 
projected in the same manner, point by point, the radius of the correspond- 
ing spherical zone from formula (1) being used as above explained. 
Derived from the above formulae, the latitude spacings on the different 
meridians are kept in nearly true distances in the central part of the 
projection. On the outskirts, the meridians and parallels increase in 
length, but the greatest distortion falls mostly beyond the limits of the 
country. 
Numerical Example. Suppose 4> = 65° and X = 40°, what are the co- 
ordinates X and 2/? 
From the table, the meridional distance between latitudes 60° and 65°, 
is ........ Mi — 346-280 statute miles 
log Mi = 2-5394274 
log 5 = 0-6989700 
1-8404574 
constant = 1-7581226 
logr = 3-5985800 
log cos 65° = 1-6259483 
log sin 40° = 1-8080675 
log sin u = 1-4340158 
u = 15°45'45"-72 = 56745" -72 
log u (in seconds) = 4-7539332 
log27r = 0-7981799 
logr = 3-5985800 
9-1506931 
log circumference (secs) = 6-1126050 
log {u in miles) = log Wm = 3-0380881 
Um = 1091 • 662 miles 
