84 A. Mukhopadliyay — Differential Equations of Trajectories. [No. 1, 

 whence 



cos 



6 

 cos- 



The equations 



X = ^P (l+COS e)=2 yp COS'^- 



Y = 4/ sin ^ = 2 \|/ sin - cos - 



which give the coordinates of any point on the trajectory, therefore, 

 become 



X = 2X cos - cos 



Y = 2A sin - cos 

 2i 



(-0- 



Since 



X3 + Y2 = 4XScos2 (a- I) 



it is easily shewn that the trajectory is the circle 

 ajS + ^z^ z= 2X (£(? cos a-f 7/ sin a). 



Example VI. — To find the oblique trajectory of a system of para- 

 bolas which have a common focus and principal axis, and whose equation 

 is, therefore, obtained by varying m in 



2/* = 4m {x-\-n%). 

 Putting m = a^, 



any point on the curve is seen to be given by 



a; = ^3 _ ^2, 

 y = 2ae. 

 The coordinates of any point on the trajectory are, therefore, given by 



Y = 2 0rp, 

 where »|/ is to be determined as a function of $ from the System 



A =20+, 

 BO that we have 



