KINETICS OF A PARTICLE. 



has the general integral 

 a = 



cos 



u', or u* = cos (6 a), 



where a is the constant of integration. The orbit has, therefore, the 

 equation 



c 1 



(16) 



which agrees in form with the equation (74) given in kinematics (Part 

 I., Art. 242), excepting the different notation used for the constants. 



132, The equation (16) represents a conic section referred to its 

 focus as origin. The general focal equation of a conic is 



lie 



- = -7 + -, cos (0 a), (17) 



r I I 



where / is the semi-latus rectum, or parameter, e the eccentricity, and 

 a. the angle made with the polar axis by the line joining the focus to the 

 nearest vertex. 



In a planetary orbit (Fig. 19), the sun S being at one of the foci, 

 the nearest vertex A is called the perihelion, the other vertex A' the 

 aphelion, and the angle a made by any radius vector SP= r with 

 the perihelion distance SA is called the true anomaly. 



Comparing equations (17) and 

 (16), we find, for the determina- 

 tion of the constants : 



Fig. 19. 



hence, 



or, solving for c and h, 



<r = V/I7, h = p.j 



(i9) 



133. The expression for the eccentricity e in (18) determines the 

 nature of the conic ; the orbit is an ellipse, parabola, or hyperbola, 



according as *=i; hence, by (18), according as the constant h of 



