1887.] A. Mukhopadliyay — Differential Equation of a Trajectory. 117 



G. signatce Stol. sat affinis differt corpore in parte granuloso (in 

 G. signata omnino granulosa) et scuto abdominali flavo quadrivittato 

 (in G. signata tantum flavo-marginato). 



IX. — On the Differential Equation of a Trajectory. — By Asutosh Mukho- 

 PADHYAT, M. A., F. R. A. S., F. R. S. E. Communicated hy The 

 Hon'ble Mahendralal Sarkar, M. D., C. I. E. 



[Received April 28th ;— Read May 4th, 1887.] 



§ 1. The problem of determining the oblique trajectory of a 

 system of confocal ellipses, appears to have been first solved by the 

 Italian Mathematician Mainardi, in a memoir in the Annali di Scienze 

 MatJiematische e Fisiche, t. I, page 251, which has been reproduced by 

 Boole (Differential Equations, 4th edition, pp. 248-251). Representing 

 half the distance between the foci by A, and the tangent of the angle 

 of intersection by n, we obtain for the equation of the trajectory, 



where C is the constant of integration, and M is given by the quadratic 



(x^Ji.y^ + h^)M. = x (M^-^-h^) (2) 



Now, this form of the equation is so complicated that it would be 

 a hopeless task to have to trace the curve from it ; indeed, it is so 

 unsymmetrical and inelegant that Professor Forsyth in his splendid 

 work on Differential Equations (page 131) does not at all give the 

 answer. In the present note, the curve is represented by a pair of 

 remarkably simple equations which admit of an interesting geometrical 

 interpretation. 



§ 2. Assume then 



xM = h^co8^<f> 

 C = 2n\ 

 where A is a new constant. Substituting in (1), we have 



log ^ = 2n\ + 2n<l>. 



/i 'M 



