1888.] A. Mukhopadhjay — Differential Equations of Trajectories. 97 



or 







l4.e2^(p + </>)p' 



whence 





We have, therefore, the 

 If we put 



h cosh ^(j9 + </>). 

 i system 



\ = li cosh n{p + <^) 

 fx = h cos <^. 



system obtained above, viz., 

 X = A cosh 6 



this is equivalent to the 





fxzzJi cos - {B —pn). 



If we had used for 

 we should have to put 



M the value 



M =z — , 



Mx = X2, 



M /i2 



~— ..2' 



03 /U,* 



whicli shews that X, /a would be interchanged in the equation of the 

 trajectory, viz., that would give the system 



X=zh cos (p, 



fjL = h cosh n(p-\-<p), 

 and it is important to notice that this second system does not admit of 

 being derived from the differential equation in elliptic coordinates, 



dX dfx 



v/A2 - h^ ~ y/h^ - /x2' 

 For the above system is the solution of the differential equation 



dX djx 



13 



