470 



SCIENCE. 



[N. S. Vol. XXII. No. 563. 



posite the given angle. Let h, c and A de- 

 note the given parts and C the required angle. 

 From the fundamental equations 



cos a = cos b cos c + sin & sin c cos A, ( 1 ) 

 cos c = cos a cos 6 + sin a sin 6 cos C, (2) 



sin sin a =^ sin A sin c, (3) 



the quantity a may be eliminated by first 

 dividing (1) by (3), then (2) by (3) with its 

 members interchanged, and then comparing 

 these results. After reducing to a simple 

 form, there results the well-known equation 



then 



y 



= — cot C, 



— tan 0. 



(7) 



Since we are here concerned only with the 

 ratio of x and y it is convenient to write 



X = COB A cos 6 — sin b cot c, ( 8 ) 



y = sin A, (9) 



provided cot c does not become too great, and 

 the ratio x/y or y/x will remain as before. 



center. A system of straight lines radiate 

 from the common center of the semicircular 

 arcs. The angles formed by these lines g,nd 

 the initial line are written in the margin of 

 the diagram. Although not shown in the 

 sketch, the entire diagram is covered by sys- 

 tems of horizontal and vertical lines differing 

 in color or character from the lines already 

 referred to. The entire sheet is thus divided 

 into small squares, the purpose being to en- 

 able one to work accurately even if the paper 

 should become somewhat distorted and also 

 to work without marking up the permanent 

 diagram. 



If we locate the point x, y upon the azimuth 

 diagram, then by (7) the angle at the center 

 which this point makes >with the direction 

 — a; is the angle C 



The product sin h cot c of (8) is positive 

 or negative according as c lies between 0° and 

 90° or 90° and 180°. Its numerical value is 

 the horizontal distance from the central ver- 

 tical line, measured on a level with the vertex 

 of the circle whose radius is sin h, to the radi- 

 ating line numbered c. 



Upon the radiating line which makes the 

 angle A with the initial direction, mark two 



70 90 

 40 30 20 



Fig. 1. 



The azimuth diagram or chart may be de- 

 scribed by aid of a sketch, Fig. 1. The radii 

 of the system of concentric semicircles are 

 equal to sin where varies from 0° to 90°, 

 counting from the center, or to cos $ where 

 varies from 0° to 90°, counting towards the 



points, one where it crosses the outer circle 

 and one where it crosses the circle whose 

 radius is cos 6. Follow the horizontal and 

 vertical straight lines until a point is found 

 on a level with the first point and on the 

 vertical passing through the second point. 



