98 A. Mukhopadliyay — Differential Equatio7is of Trajectories. [No. 1, 



which is different from the one given above, viz., this leads to the 

 primitive 



n cos -+cosh ■-"\rfc=zO, 

 rh h 



X = ^ cos 0, 



so that, if 



we must have 



/ji=ih cosh n(p-{-(p). 

 We see, then, that, because M is given by a quadratic, Mainardi's result 

 is really equivalent to two, viz., we have the two systems 



X=^h cosh n(p + <l>) 



fx = h cos <t>. 



X=h cos cji \ 



ix=zh cosh n(p-^<t>) ) 

 and these two systems are the solutions of the two distinct differential 

 equations 



dX du 



dX dfx 



If, now, we consider for a moment these two differential equations, we 

 see that the first belongs to the trajectory which intersects the confocal 

 ellipses at an angle a (where 7^ = tan a), while the other belongs to the 



trajector}^ which intersects the conf ocals at an angle ( h "" ^ I > ^ii 



— = tan ( -T — a )• 



n \2 / 



But, since the confocal hyperbolas intersect the ellipses orthogonally, it 

 follows at once that the second differential equation belongs to the 

 trajectory which intersects the confocal hyperbolas at an angle (tt— a), 

 in both cases measuring the angle in the same sense ; hence, the solution 



which leads to the system 



X=h cosh n(p-\-<l>) 



fji = h cos </> 

 is relevant, while the value 



TIT ^^ 



which furnishes the other system 



X = h cos (p 



fjL =zh cosh, nip-^-cfi) 



since 



