38 



II. PAETICULAR MOTIONS OF A POINT. 



the simplest case, where one frequency is twice the other, and the 

 phase difference is s, we have 



x = asin.(nt e), 

 15) 



y = 6 sin 2 nt. 



Expressing sin2nt in terms of cosnt and eliminating the functions 

 of t we obtain 



16) ^ + ^sin 2s = l 



Rationalizing this we shall obtain a curve of the fourth order having 



one double point, shown 

 in Fig. 11, for s = 0. If 



e = ~> 16) becomes 



> S+U-T' 



a parabola (Fig. 11). Since 

 we may always express 

 sinw# rationally in terms 

 of sin#, cos 'x, when m is 

 an integer, the elimination 

 may always be performed 

 and the curves will be 

 algebraic. 



. 11. 



2O. Central Forces. Having now dealt with two cases in 

 which the acceleration passes through a fixed point, - - that of the 

 motion of the planets and harmonic motions, it will be convenient 

 to treat the general case. In 12 we found the nature and magnitude 

 of the acceleration by the differentiation of the equations expressing 

 the motions. We will now consider the inverse problem, that of 

 obtaining the equations describing the motion by integration of the 

 differential equations of motion when the force is given. 



We have by 10, 34) and 35) for the radial acceleration in 

 the direction away from the center, 



18) a r = ~ - 



dv 



and for the transverse acceleration, 



If the acceleration is central a y = and we have by integration 



20) ,**?_, 



Kepler's law of areas. 



