Page 287 hydrography 371 



371. Derivation of ForjMULAS 



In the general consideration of the problem, certain fundamejilal formulas are 



necessary. These are derived as follows: 



In figure 62, B and C are control stations, 

 a = distance from B to C. 



a and -^ = observed angles from B to C. 

 A and ^' = centers of circles which are loci of all points where angles a and -^ can be observed. 



d and d' = AD and A'D respectively — measured along the perpendicular bisector of BC. 

 c=: radius of circle BA'C. 



In the circle BA'C, 



d= "H cot a. (1) 



c= -7i cosec a. (2) 



It can also be shown that, 



a a a a ,„, 



-n cosec «= "9 cot -X — n cot a. (o) 



That is, the radius c for any angle a equals the difference between the d distances for -^ and «; in 



other words, c = d' — d. Stating the relationship in another way, the arc for an angle a will pass 



through the center for the arc of angle -~. (See 3751 for application of this relationship.) 



In computing the d and c values, a scale factor is introduced so that the results can be measured 

 directly on a 1:10,000 scale meter bar. For any scale, the formula is: 



a , . , /VN 1 0,000 



Scale factor (a) = — -, 3* 



scale used 



Thus, if the scale used is 1:40,000, X=-r- 



A" 

 Figure 62. — Principle of plotting angles without a protractor. 



