94 A. Mukhopadhyay — Differential Equations of Trajectories. [No. 1, 



number of curves whose oblique trajectories may be determined with 

 ease by the theorems and methods of this paper ; but it is needless to 

 multiply instances, as the examples given above will, it is hoped, amply 

 illustrate these observations. 



I6th Novemher 1887. 



Additional Note on Mainardis Problem. 



Since the above paper was read, I have been informed by Prof. 

 Booth that Prof. Michael Roberts, in his Lectures on Differential Equa- 

 tions delivered at the University of Dublin, used to solve Mainardi's 

 problem by the help of elliptic coordinates ; I have not the opportunity 

 of examining the solution arrived at by Prof. Roberts (as I believe it 

 has never been published), but I give below the results I have obtained 

 by means of the coordinates suggested. 



If a be the semi- axis-major of the primitive conic, and Ji half the 

 distance between its foci, its equation is 



a^ a^-h^~ ' 

 and any member of the confocal family is obtained by varying a ; so 

 that, if Xt H- ^6 ^^® elliptic coordinates of any point P on the trajectory, 

 they are determined from the system 



viz.f A. is the semi-axis-major of the ellipse, and ft the semi-axis-trans- 

 verse of the hyperbola through P confocal to the primitive one ; hence, 

 solving between these equations, we have 



y- p • 



Taking the logarithmic differential, we see that the element of arc of 

 any curve through P is 



de^ = A^-^dy^ = ^, dX^ - -^^ dp^. 



Hence, if ds-^, ds^ be the elements of arc of the confocal ellipse and 

 hyperbola whose semiaxes are A, fi, and which intersect orthogonally at 

 P, we have, for the ellipse regarding X as constant, 



X2-/x2 ^ ^ 



a;» = 



