1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 95 

 and, for the hyperbola regarding /x as constant, 



Now, if a be the angle of the trajectory, viz,^ the angle at P between 



the trajectory and the ellipse (X), we have clearly 



dso 



-— ^ = tan a = n. 



dsi 



Hence 



or 



dX djuL 



Integrating, we have 



log (A+ v^X23l2) = - w cos "^ ^+ h 



which is, accordingly, the equation of the trajectory in elliptic coordi- 

 nates. It will be remarked that, though the application of elliptic 

 coordinates removes the difficulties of integration, the result is not 

 obtained in an appreciably simpler form ; and, besides, the method is 

 not one of general application, as it requires a knowledge of the elements 

 of arc, as well of the given curve as of its orthogonal trajectory ; the 

 methods and theorems of this paper, however, effectually remove these 

 disad vantages. 



It may usefully be noted that if we use the inverse hyperbolic 

 functions, the integral of 



d\ da 



may be written 



-1 X _i M 



cosh 7- -|- n COS T=^j 



h h 



and this at once shews that if we have 



Xz=h cosh 0, 

 where ^ is a variable parameter, we must have 



fi = Ji cos - (k — 0). 



In this form it is not difficult to identify our solution with Mainardi's 

 result, viz.} 



