86 A. Mukhopadhyay — Diferential Equations of Trajectories. [No. 1, 



df2 -^ 







dO' 



-^+^2 







^, 



* 







d^ ' 



■6H^2 







d^ 



e^ 





"6H^2' 



whence 











L = 



1, M: 



= 0, N= -1, 



and the differential 



equation for 



^ is 







dy\) 









dO'' 



= -w, 



which gives 



rp = n{\- 6). 

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



X = log a \/e^-^n^ (A-^)2 



d 



It can easily be shewn from this system that the actual equation of the 

 trajectory is 



e (sin y-\-n cos y) = a\n, 

 or, if a be the angle of the trajectory, this becomes 

 e^ sin (2/4-a) = a\ sin a. 

 §. 5. Conjugate Functions. 

 It will be remarked that in some of the examples given above, the 

 integration of the differential equation for \p was materially facilitated 

 whenever we found 



M = 0, L= +N. 

 It is, therefore, a matter of importance to discover under what circum- 

 stances this may be expected to happen. 



Theorem. — The coordinates of any point on a curve being given by 

 a?=/i(6l, a), 



and, the coordinates of the corresponding point on the trajectory by 



x=/i(e,F, (+, A)], 



if we have 

 and 



dO d^ "^ d6 d^ " ' 



to prove that /^ and /g must be conjugate functions of i}/ and 0. 



