1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 81 



so that the coordinates of any point on the trajectory are 



X = 6, 



where, to determine i^ as a function of Of we have 



which furnish the system 



&-1 ^-^ 



dij;-^' #-^' 

 and by virtue of these, we have 



L =1 + 1/^2, 



M = OiJ/, 



whence, the differential equation for if/ is 



#_ nh _ n (1+>A^) 

 dO "" N-^M ~ -e-nOiP 

 which gives 



l+w»A ,. dS 



Integrating, we get 

 tan" 

 which easily reduces to 



tan-' ^^ ^log (1 + ^2) = -71 log -, 



- tan ' \j/ 



V^l + ^s 

 Hence, finally, the coordinates of any point on the trajectory are given by 



\ tan"' ij/ 



■\/l + «/'^ 







v/l + ^2 



-- tan~'0 

 n 







It is 



5 not difficult to shew that these values lead to a 



well-known 



result ; 



for 

 and 



we have 



X ^ 



(XHY^y=Xe 



-- tan"' \Jf 

 n 







11 



