1888.] A. Mukhopadhjay — Differential Equations of Trajectories. 75 



which lead to the values 



dy c?X dx dT 

 de dO^dO lO 

 ^chpi^df,_df^ df,) 

 dO \ dxb dO dO di}/ ) 



dx dX. dy dY 



dxlf dO dij/ 



^\dd) ^\doJ '^ dO {le 



Hence, putting 



-(3)V(sr <») 



dO dil^^ dO dxl;' ^ ^ 



c74 dfj^_dfi df^ 



~ do # do #' ^ ^ 



we have finally, from (7), the equation 



d_l _nlj_ 



dO N-^M' ^ ^ 



which is exactly the theorem enunciated above. 



It may not be altogether unprofitable to note that the trajectory is 

 determined by two conditions, viz., in the first place, we have to vary 

 the constants in a definite manner ; and, in the second place, the trajectory 

 is to intersect the curve at a given angle ; the first of these conditions 

 leads to the actual values of the coordinates of any point on the trajec- 

 tory, furnished by (5) and (6), while the second condition determines 

 the relation between $ and xp which enter into the values of those coor- 

 dinates. 



§. 3. Application to Mainardis Problem. 

 Example I. — In order to test the power and generality of this 

 theorem, we shall apply it to solve Mainardi's problem of determining 

 the oblique trajectory of a system of confocal ellipses. The primitive 

 ellipse being 



i4=^' (-) 



we get the confocal system by varying a and h subject to the condition 



a^-h^ = h^ (18) 



The coordinates of any point on the ellipse are given by 



x=ia cos 6, 

 y = h sin 0, 



