1888.] A. Mukhopadhjay — Differential Equations of Trajectories. 91 

 Example IX. — Take, again, the curve 



and suppose that its oblique trajectory is required, when a and h are 

 made to vary subject to the condition 



a^ - 62 = Ifi. 

 The coordinates of any point on the curve may be written 



a cos 6 



aj = 



a^cos2^ + 62sin2 $' 



h sin $ 

 a^ cos2^+&2sin2^' 



and we have also 



azzh cosh ^ 

 h — li sinh ^. 

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

 __ cos 6 cosh vj/ 



" h (cos2 6 cosh2 ij/ -l-sin2 Q sinh^ ^) 

 _ 2 cos 6 cosh xj/ 

 ~ 7i (cosh 2t|/ + cos 2^) 

 __ sin ^ sinh \|/ 



"" /i (cos2 6 cosh2 4/4-sin2 ^ sinli^ ^) 



2 sin ^ sinh \|/ 

 ~ /^(cosh 2;j/+cos 2^)* 

 To determine \(/ as a function of 6, we have 



But /i and /2 are two conjugate functions ; for we know that if we 

 separate the real and imaginary parts of 



sec (a + \/'=n^ ^) = A + v/^ B, 

 we have 



. __ 2 cos a cosh /3 



B = 



cosh 2;8 -j- cos 2a 

 2 sin a sinh y8 



cosh 2;8+ cos 2a' 

 Hence, by the theorem of this section, we have 



L + N = 0, M = 0, 

 and the differential equation for f becomes 



whence \|/=n(A— 6). 



