1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 93 



and, as before, 



^ = n(\'-6). 

 Therefore, we finally infer tliat the coordinates of any point on the 

 oblique trajectory of the curve 



tan^ y = -X- , 



when a and h vary subject to the relation 



are given by the system 



26^^ = cosh 2n (X-O)- cos 20. 

 tan y = tanh n(\ — 6). cot 6. 

 From the theorem established in this section, it is again evident 

 that, if h{0,^), f 2 (0,^1^) 



be any two conjugate functions of 6 and xf/, and the equation of a curve 

 be obtained by eliminating from the system 



the equation of the oblique trajectory of this curve when a is made to 

 vary is obtained by eliminating 6 from the system 



x=/i [e,n(\-e)] 



Similarly, if the equation of a curve is obtained by eliminating i/' from 

 the system 



the equation of the oblique trajectory of this curve when a is made to 

 vary is obtained by eliminating i// from the system 



Y = /2 [n{X + ^), ^]. 

 Again as from the well known formula for expanding 



and separating its real and imaginary parts, viz., 



we can determine at pleasure an infinite number of pairs of conjugate 

 functions, it is clear that we may obtain without any difficulty an infinite 



