1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 77 



The coordinates of any point on tlie hyperbola are given by 



x=za cosh 0, 

 y — h sinh 6, 

 where a = Ji cos if/, 



b = Ji sin if/j 

 so that the coordinates of any point on the trajectory are given by 



X = A- cosh 6 cos xj/j 

 Y = 7^ sinh sin if/. 

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



f-^=h cosh cos if/, 

 f^ = h sinh 6 sin \f/, 



whence 



1 /? 



-^ = h sinh ^ cos j/', 

 du 



■~~ =Ji cosh ^ sin if/, 

 du 



-~r = — h cosh 6 sin i/^, 



"vf = ^ sinh ^ cos if/, 

 dif/ 



and, therefore, 



L = h^ [sinh2 6 cos^ .// + cosh' ^ sin^ ,/.} 



M = 



N = - /i2 1^ sinh2 ^ cos2 (/.+ cosh^ e sin^ ./. J 



The differential equation (11) becomes 



dif/ 



so that if/ = n(\ — 6), 



where, of course, X is a constant different from the X in the solution of 

 Mainardi's problem. The coordinates of any point on the oblique 

 trajectory of a system of confocal hyperbolas are, therefore, given by 



'K = h cosh 0. cos n {\— 6). 



Y=/isinh 6. sin7^ {X-6). 

 If we put 



6 = \ — , \ n= — a, 



n 



these equations may be written 



X= /i cos <?>. cosh - (/x-f</>), 



Y = — A sin </>. sinh - (fx \ <p), 

 n 



