118 A. Mxikho^ndhjaj—BifferenHal "Equation of a Trajectory. [No. 1, 



M 





1+ 



V a? 



^l i_j.e«<^ + <^) 



. V X 



i — e 



^_M^ (l_e^H^-^^))^ 





aj I ^ 2n(A + ^)j ~ I 1 . M^ + 1>)\^ 



But, M = — cos*d» 



a? 



.'. Substituting in the above and extracting the square root, we get 

 h 26"'"^*' 



- COS <^ = , . . > 



,'. x=.h cos <i) ,, , .. ■ 



= ^cos<^i|e^ ^^ + e V r/ J 



= ^ COS ^. cosh w(X4-^)« 



Again, substituting the value of M in (2>, we have 



^ , h^ cos''<l> /h* cos''((> , , \ 

 (:.^^y^ + h^) ^=^[ '^-^-+^7 



.'. x^-\-y^ + h^ = h^ cos''<l>-\-x'' sec^<t* 

 .'. y^ -\-h^ sin2(^ _ ^2 tan^i^ 



. ^l ^'_= 1 



* * h^ cos2<^ h^ sin^^ ' 



and, since we have shewn that 



x = h cos <t>. cosh 7i(X + <^), 

 we see at once that 



y ^h sin <p . sinh n{\-\-<p). 

 Therefore, the co-ordinates of any point on the trajectory may be 



