1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 83 



and it is not difficult to sliew that these values belong to the logarith- 

 mic spiral. 



Example V. — To find the oblique trajectory of a system of circles 

 which touch a given straight line at a given point, and whose equation 

 is, therefore, obtained by varying r in 



x^-\-y'^ = 2rx. 

 The coordinates of any point on the circle are given by 

 x=.r (14-cos 6), 

 y — r sin $, 

 so that, the coordinates of any point on the trajectory are given by 



X = ;|/ (l + cos 6), 

 Y = ^ sin 6. 

 To determine f as a function of 0, we have 



/,=^|/(l+COS^), 



/a = ^ sin e, 



which lead to the system 



d^ 



whence, we have 



M = - ij/ sin ^ 

 N=r|, (l+cos 6). 

 The differential equation for v|/ reduces to 



^v}/ _ n y\i 



dd 1 + cos 0-\-n sin 0' 



Writing n = tan a, where a is the angle of the trajectory, we have 



d^, . die- a) 



= sm a ^ ^ 



\|/ cos a + cos (^ — a)' 



Integrating, we have at once 



a . a 0- 



cos- + sm-tan — 



log - = log 



A "^ a . a 



cos — — sin - tan 



2 2 2 



