1888.] A. Mukhopadhjay — Differential Equations of Trajectories. 73 



Mainardi, by no means less complicated tlian his result, could be materi- 

 ally simplified. I was, thus, led to the following very general theorem, 

 which it is my object in the present paper to establish and illustrate, 

 and, which shews that whenever the coordinates of any point on a curve 

 can be expressed by means of a single variable parameter, the coordinates 

 of the corresponding point on the trajectory may be similarly expressed; 

 and, as an immediate corollary to my theorem, I have pointed out the 

 relation which connects it with the theory of Conjugate Functions.* 



§. 2. Theorem. 

 Theorem. — If the coordinates of any point on a curve are expressed 

 by means of a variable parameter ^, by the two equations 



^ = /i (6, a), 



where a and h are two arbitrary constants ; and, if we seek the oblique 

 trajectory of the system of curves obtained by varying a and b, subject 

 to any condition which can be analytically represented by means of a 

 parameter if/, as equivalent to the system 



where ^ is a known constant ; the coordinates of the corresponding point 

 on the trajectory are given by the system 



where \f/ is given as a function of by the di:£erential equation 



cl\J/ n L 



where n z= tan a, 



a being the angle of intersection of the curve and the trajectory, and 



dO dxp^ ae # 



d6 dip dO dip' 

 To establish this theorem, let us first fix the ideas by confining our 

 attention to one definite member of the given family of curves as well as 

 to one of the trajectories ; then it is clear that the common point of inter- 

 section of the curve and the trajectory, may be arbitrarily regarded as a 



* For a fall analysis of this paper, see the Froceediniif< for 1887, pp. 250-251. 

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