1888.] A. Mukliopadhyay — Differential Equations of Trajectories. 79 



As the two constants of the general theorem are here equal, the coordi- 

 nates of any point on the trajectory are given by 



X = i/' tan2 e 



Y = 2 i/^ tan ^. 

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



f^zzxp tan2 6 



/2 = 2 V' tan e, 

 which give 



^ = 2 ./^ tan ^ sec' 0, 





de ' 



= 2xp sec2 (9, 







dlf' 



= tan2 e 





so that we have 



dxp 



- 2 tan (9, 







Ij = 4<ip' 



2 sec^ (9 







M = 2«/. 



tan(9sec2(9 (2 + tan3^) 







N= -^ 



J xp tau2 ^ sec** (9 





and the differential 



equation 



for \\i becomes 





dxp 





2n i/' sec'*' B 





dS 



~ tan {2n + tan O-^-n tan^ 0) ' 





This may be written 







dxp 





2n sec* (9 tZ ^ 





^ 



tan 6 (2n-\- tan e-\-7i tan^ Oy 





which, by putting tan 6z=.z, 



reduces to 







dxp 

 •A" 



« {2n-\-z-\-nz^) 





or, 









dxp 3 dz 



dz l{2nz^-l)dz 





J-\ 



l2n-{-z + 



nz^ z 2 2n-^z-\-nz'^ 





Integrating, we hav< 



3 



'2>v/r= 







log^ 



2nz^l-\/l- 



Sn^ 



■ 8^2 " 2n2; + H-V'l- 



8ti2 





-log 2; 



-ilog (2n + z-^nz^), 





which gives 



X f 27^tan^ + l-v/l- 8nO 2 \/l - 87^2. 



''^ "" tan ^ -v/(2»* + tan ^+ w tan^ $) \2n tan ^ + 1 -|- \/l - Bv^^j 



