82 A. Mukliopadhyay — Differential Equations of Trajectories. [No. 1, 



Transforming to polar coordinates, by putting 



X = r cos </>, Y = r sin </>, 

 we have 



tan ^ = i/' 



tan~^ \p 



wlience, 



r = A e 



r = A e , 



whicli is the logarithmic spiral. 

 Second method. 



We might also have proceeded as follows, viz., putting a = tan )8, 

 the coordinates of any point on the line are given by 



xzz. e^ cos /?, 

 y= e^ sin p. 

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



X = e^ cos j/', 

 Y z= e^ sin xl/. 

 To determine i/' as a function of 6, we have 



/i = e^ cos j/', 

 /a = e^ sin ,/., 

 whence, we have the system 



^/i 6 



§ = /sin^, 

 |=-/sin,, 



| = e^os,, 

 which furnish us with the values 



The differential equation for if/ becomes 



d^\f 



whence ^\l =n (X — 6). 



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



X = e^ cos n{\-6), 

 Y = e^sinw(\-^), 



