170 TT. S. COAST AND GEODETIC SURVEY. 



•is measured to the right and left of QQ' and r] is measured 

 up and down from P^P\ 



In the figure 48 let P he the pole and let RBR' be the 

 Equator and also let ABA' be the great circle that we 

 wish to make correspond to the central meridian of the 

 ordinary projection. BR and BA are quadrants, and AR 

 measures the inclination of the given great circle to the 

 plane of the Equator, and PMA becomes the Equator on 

 the transverse projection. Let Q be the intersection that 

 we wish to compute. We have 5^ = 90°-;/'; QP = 90°-.^; 

 BP = 90°; lBPQ = 90°-\; ZABR = ^; ZP^9 = 90°- 

 (/3 + t;). By the trigonometry of the spherical triangle we 

 obtain from these results the relations 



sin \f/ = sin X cos (p 



cos ^p cos (J3 +7]) =cos X cos (p 



cos \l/ sin (/3 +77) =sin (f, 



or by combining the last two equations 



tan (jS + r?) =sec X tan <p, 



j8 is a constant the value of which is known from our choice 

 of the great circle that is to form the center of the map; 

 it is the value of the parallel of latitude to which the great 

 circle is tangent. 



By use of the equations 



sin ^ = sin X cos cp 

 and 



tan (/3 + ?/) = sec X tan (p 



we can compute the \^ and rj values for any intersections of 

 the parallels and meridians that we may wish to determine. 

 The points are then plotted on the projection as originady 

 constructed; a smooth curve drawn through the points 

 corresponding to a constant value of (p will represent the 

 parallel of latitude cp, and, similarly, the smooth curve 

 through the points corresponding to a constant value of X 

 will represent the meridian of longitude X. After these 

 curves are drawn, the original projection lines can be 

 erased, and then only the meridians and parallels will 

 appear on the projection. The folding plate represents 

 such a projection of the North Pacific Ocean, showing 

 the eastern coast of Asia in its relation to North America. 



