78 A. Mukhopadhyay— DiJere?ii5mZ Equatioyis of Trajectories. [No. 1, 



which system is slightly different from what has been obtained above as 

 the solution of Mainardi's problem ; but the equations are obviously 

 capable of a geometrical interpretation closely analogous to what is given 

 in my former paper. 



If we had to obtain by the ordinary method the oblique trajectory 

 of a system of confocal hyperbolas, we should have to eliminate a and h 

 from the equations 



nQ% ,11% 



-,-l^ = l,a' + l^ = k., 

 dy a^ 62 



t-=:p= • 



ax nx y 



The result may be expressed in the form 



{{nx-y)+{x-ny)'p^ [{x -\ ny) -\- {nx ^ y) p^ 

 = h^ (?^^-^) {\-\-np). 

 But, it is surely no agreeable task to have to find the actual equation of 

 the trajectory by integrating this differential equation. 



Assuming the expressions for the coordinates' of any point on the 

 oblique trajectory of a system of confocal ellipses, it is easy to write 

 down the expressions for the coordinates of any point on the oblique 

 trajectory of a system of confocal hyperbolas. Consider the point of 

 intersection of an ellipse and its trajectory, and draw through this point 

 the confocal hyperbola ; then, since the ellipse and hyperbola cat each 

 other orthogonally, the trajectory, which intersects the ellipse at an 



angle a, will intersect the hyperbola at an angle ( ^ + " )' i^ ^o^h cases 



measuring the angle of intersection in the same sense ; the trajectory, 

 therefore, is also the oblique trajectory of the confocal hyperbolas (at an 



angle - + a), and the coordinates of any point on it, as such, will, 



therefore, be obtained by writing for n {= tan a), j = tan -4- a ) 



Example III. — To find the oblique trajectory of a system of para- 

 bolas which have a common principal axis and which touch each other at 

 their common vertex, and, the equations of which are, accordingly, 

 obtained by varying a in 



y^ = ^ax. 

 The coordinates of any point on the curve are given by 



x = a tan^ $, 

 y = 2a tan 0. 



