MARCQ ST. HILAIRE'S METHOD (PART 2) FINAL POSITION BY COMPUTATION CALCULATING THE INTERSECTION POINT OF 1. Plotting Versus Calculating the Final Fix.-So far, the intersection of the position lines established by St. Hilaire's method has been found by plotting. This is the most logical method of finding the final fix because the position lines, obtained by whatever method of computation, are intended to be plotted on the chart. Some authorities on navigation and men of high professional standing, however, claim that the point of intersection of two position lines should be calculated as well as plotted; in other words, they maintain that a navigator ought to know how to plot and how to compute his final fix. 2. Several methods of calculating the point of intersection have been proposed and all are based on the solution by trigonometric formulas of two or more triangles formed by the intercepts, the azimuths, and the position lines. These methods are necessarily more or less complicated even though tables have been constructed to facilitate calculations and simplify the work. 3. Formulas for Calculating the Point of Intersection. One of the simplest, and perhaps the most convenient, methods of calculating the point of intersection of COPYRIGHTED BY INTERNATIONAL TEXTBOOK COMPANY. ALL RIGHTS RESERVED § 18 position lines found by the St. Hilaire method is to determine the difference of latitude and the difference of longitude between the assumed, or dead-reckoning, position and the final fix. Thus, in Fig. 1, if the angles a1 and a represent the azimuths and h1 and h2 the corresponding intercepts, the point of intersection Z of the two lines of position EZ and CZ is found by determining BD, the difference of latitude, and DZ, the departure. To find B D and DZ from the known quantities a1, a2 and h1, h2, it is convenient to make use of an auxiliary triangle RZF formed by extending the position line CZ to the meridian BF. It will be noticed that if, in this auxiliary triangle, a perpendicular is drawn from Z toward the meridian, angle RZ D=a2 and angle DZF=a1, from which follows that the sum of these angles or RZF is equal to the angle A, or the sum of azimuths. 4. Derivation of Formulas.-The procedure of finding the difference of latitude and departure may now be accomplished as follows: In the right triangle B C F, Fig. 1, B F=h1 sec a1. In the triangle R Z F, RZ: FR sin R F Z: sin FZR Now, the angle RF Z= (90°-a1) and the angle FZR =(a2+a1)=A. Therefore, Substituting in this equation the value of FR previously obtained, To find the departure D Z, a perpendicular is dropped from Z toward RF. In the right triangle RD Z, DZ RZ cos az Substituting in this equation the value of R Z found above, cos a2 (h1-h2 sec a2 cos a1) DZ= But, cos aXsec a=1; hence, whence, DZ=2 sin A sin A D Z=cosec A (h1 cos α-h2 cos a1) To find the difference of latitude BD, Fig. 1, in the right triangles BER and RD Z, or, BR=h2 sec a2 RD=R Z sin a2 BR+RD=h2 sec a2+R Z sin ag BD=h2 sec a2+R Z sin a2 Substituting the value of R Z previously found, BD h2 sec a2+ (h1-h2 sec a2 cos a1) sin a2 Multiplying, the equation becomes BD h2 sec a2+ sin A h1 sin a2-h2 tan a2 cos ai Placing the terms within the parenthesis under a common denominator, the equation becomes Again, sin (a+a2)=sin a, cos a2+cos a sin a2; hence, BD=h sin a1 cos a2+sin a2 cos a1— sin a1⁄2 cos a1, h1 sin a1⁄2 cos a2 sin A 5. The formulas found may now be used in calculating the difference of latitude and departure of the point of intersection of any two lines of position computed by the St. Hilaire method. Assuming that a, and h1 represent, respectively, the |