1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 87 



To establish this, 



we see that the conditions given, viz., 





^ = 7.(X-i-^),M = 0, 



reduce the differential 



equation 





d^\) nil 





de~:^-nM. 



to the condition 







L = N. 



Now, since 



M = 0, 



we have 





^1 . ^ 



d6 dy\f 



d6 drp 



Substituting in the value for N", we get 



dO dy\, dO dyp 



^^^^ d^ dd 



d^ 



de 



and, since N = L, 



we must have 



%. _ %^ 

 d^ dO' 



Therefore 



\d6) de d^ 



whence 



(16) 



(17) 



dO d^ 



The two equations marked (16) and (17) make it manifest that/^ and/^ 

 must be conjugate functions of xp and 0. 



In Mainardi's problem, which is the first example given above, we 

 Lave 4; = 7i (X + 6), M = 0, 



so that the quantities 



h cos 6 cosh \|/, /i sin ^ sinh ^ 

 are conjugate functions of ij/ and 6 j hence, we infer from a well-known 



