of 9- , four values of r- from (9) and four sets of rectangular coordinates from Xj = c + r-' cos 6- , 

 yj = rj' sin e. (i= 1,2,3,4) (12) 



and for each point of intersection two of the additional distances 



rj = rj' ± 2b, rj _^ 4 = rj' + 2a (i = 1,2,3,4). (13) 



A procedure for using ecpiations (9) through (13) will be described and used for two 

 examples. Since a,b,c,d,a will be given, first compute K = (d^ - b^)/(c^ - a^), fi = d cos a, 

 V = d sin a, tan /3 = v/ (/x + cK). 



From tan /S, using tables, find |S and sin |S . Then compute 

 cos yj = (± aK + b) sin /8/i/ (i = 1,2,3,4), and 



^i = ^ + ^i (i = 1,2,3,4). Next compute 



c' -a' d'- h' 



rj'= = i = 1,2,3,4 



+ a-c cos 0- d cos i6-—a)±h 



choosing the proper value (with respect to sign) of ± a, + b in each member which will make 



them equal and positive for each value of d-. Now the rectangular coordinates may be computed 



from X- = c + r-'cos 6-, y- = r-' sin d- . Useful checks are provided at this point by the relations 



(x. - c)' + y.^ = rj'^ and by Ixj =- H from equation (3). H = 2a' (Dj8„ - S)/L, /3„ = CG+Aa\ 



8 = BEG, L = )So' - GB'a% G = c^ - a\ A = ^l' -h\ B = 2fiv, C = i/' - b\ D = 2(r// - cA), 



E = 2Si/, r = b' - d', S = r - c^. Finally compute the additional distances r- = r-' ± 2b, 



rj + 4 = rj' ±2a. (i = 1,2,3,4). 



Example 1. Let c = d = 2, a=b=l, a = 45°. sin a = cos a = \/2/2. 



K = (d' - b=')/(c=' - a') = 1. v = ii=2 (0.70710678) = 1.41421356. 



tan /3= v/{ii + cK) = (1.41421356)/(3.41421356) = 0.41421356. 



/8 = 22°30' , sin /3 = 0.38268343. 



cos yj = (+ aK ± b) (sin ^/v) = (± 1 + 1) (0.27059805) = ± (0.54119610), 0. 



< y; < 277. 



yj = 57° 14' 05':666, 90°, 122° 45' 54';334, 270° 



(9j = ^ + yj, e, = 79° 44' 05'.666, O^, = 112° 30' , d, = 145° 15' 54'1334, 6, = 292° 30' 



3 3 

 J.' = = . (Choose the proper value of ± 1 in each member which 



1 ± 1 - 2 cos 0j 2 cos (^j - 45) ± 1 



will make them equal and positive for each value of 6-^. If this cannot be done the values of 

 6- may be in error.) The work may be arranged in table form as follows: 



105 



