Dmputed from (5), and an arc of radius l/(X.g) is drawn from 

 0, p = 1 on the p , s-diagram. The center of °- curvature lies 



is then con 



point s = 0, p = 1 on the p , s-diagram. The center 5f"- curvature lies 

 on a line perpendicular to the known tangent direction, tan 9 = B^/Ds = 

 i.e., on the p-axis. 



The distance s = s^ along the ray to the next contour is obtained, 

 approximately, by measuring the length of the chord or tangents drawn in 

 the ray construction. The circular arc in the p, s-diagram is extended to 

 s = S-, and the value p = p. is read. The angle 9=9 is obtained from 

 the dxrection of the tangent at s = s^, p = p^, and then (Kq)-, is computed 

 from (?) introducing p , 9-, and the values of p and q appropriate to the 

 point s = s^ on the ray, A second arc of radius !/()( )^ is smoothly 

 joined to the first by taking the center of curvature on a line perpendicular 

 to the tangent at s = s^, p = p^. The arc is extended to s = s_, the 



distance along the ray to the next contour intersection, and the process 

 is repeated. The integral curve p = p(s) for (l) is thus approximated. 



The application of Kelvin's method in solving (1) parallels the use 

 of the method as a basis for the direct construction of the ray (Arthur, 

 Mmik, and Isaacs, 19^2), A3 in ray construction, the radii of curvature 

 may be large, and it is more suitable to sinply construct the chord or 

 equal tangents to the circular arc as sho^m to the right in Fig, k. The 

 angular change A© , which is required for the construction, is calculated 

 from 



sin(9 + Ae) = Kg .As + sin Q , (7) 



a relationship wMch follows from the geometry of the figure. 



For example, knowing 9^ and (Ko)]_ at s = s-j^ in Fig, It, the angular 

 change A9^ for the interval As^ ~ ^2 ~ ^1 ^^ calculable from (7) written 

 in the form 



A9^ = sin'-"- [(Kg);,^ . As^ + sin 9^1 - 9^ . 



The chord or equal tangents are constructed from s^, p and their inter- 

 section with the line s = s gives p = p , The integral curve 



at Sg, P2 makes an angle ^2 '^ ®1 "^ ^^1 ^^^ "^^ direction of the s-axis. 



If desired, the grsphical construction can be avoided and the change 

 Ap calculated from 



Ap = As . tan (9 + ^ ) (8) 



Equations (7) and (8) are enployed in the example which folloi^s, 



-7- 



