1 48 



UNIVERSITY OF COLORADO STUDIES 



The meaning of equations (30) and (31) is clear: The first represents 

 a straight line through the point (a, 0) and the origin ; the second a circle 

 with the point (a/2, b/2) as a center, and OM as a diameter, Fig. 4. 



This result is also obtained from the expressions (25) and (26). For 

 k = 00 the first indicates that the center of the circle (3) is at an infinite 



Fig. 4 



distance in a direction whose trigonometric tangent is 



U + ak + aVi + k 2 ) a 



lmW ( = • 



(a — bk—bVi + k 2 ) k=^ccb 



The second expression gives for the co-ordinates of the circle (24) 

 x = a/2, y = b/2 . 



IV. A COMPOUND CURVE PROBLEM 



As an application of the preceding theory I shall solve the following 

 well-known problem: 



Given two tangents T 1 and T 2 of unequal length, their angle 0} inter- 

 section (f>, and one radius R t , to find the length of the radius R 2 of the 

 other branch of a compound curve which will unite the two tangents. 



