7'i A. Mukliopadbyay — Differential 'Equations of Trajectories. [No. 1, 



point, either on the one, or on the other ; and, from each point of view, 

 the coordinates satisfy two entirely different equations, though their 

 actual values are the same in both cases ; hence, if the coordinates of the 

 point, regarded as a point on the curve, be furnished by the system 



ie^/iC^, a), (1) 



y=h{e,h), (2) 



and the trajectory is obtained by varying a and h subject to the limi- 

 tations 



a = ri(iA, /.), (3) 



h^Y^i^p.h), (4) 



the coordinates of the corresponding point on the trajectory must be 

 obtained by substituting in (1) and (2) the values of a and h from (3) 

 and (4), viz^ we have 



X=A {^,Fi(^,/.)} (5) 



Y=/2 [6,Y, {^p,h)} (6) 



In the next place, we have to determine i// as a function of 6, and this is 

 easily obtained from the condition that the trajectory intersects the curve 

 at a constant angle a. Now, it is well-known that 



Tx dX 

 are the trigonometrical tangents of the angles which the tangents to the 

 curve and to the trajectory, at their common point of intersection, make 

 with the axis of a ; hence, ii n = tan a, we have 



dy_dY 

 dx dX. 



dx tZX 

 dy dX dx dY 



_ ~d6 le'le dO 



~dxdX dy_dY ^^ 



dO dO '^ dO dO 

 Remembering that in differentiating X and Y with respect to $, we must 

 regard ^ as a function of ij/, but not so in the case of x and y, we have 



^ _^ dy_ df^ 



dO " dO* dO~ dO 



dX_dfj^ df^dj^ 



dO "dOdil^dO' 



dY_dJ^df^dAJ^ 



de" dO dxp dO ' 



