80 A. Mukliopadhyay — Differential Equations of Trajectories. [No. 1, 

 This holds so long as ^n^ Z. 1, or, if a be the angle of the trajectory 



If tan a be greater than this value, the corresponding value of x^/ will be 

 still more complex, but may easily be found. In the particular case 

 where 



the differential equation for xp reduces to 



"i/T" ^ {z-\-^2)'^~"7 z-\-,/2 

 Integrating and substituting for z, we have 



-3 v/2 



^ tan e (tan + ^2) = e ^^^ ^ + v/'^. 

 If the orthogonal trajectory be required, the expression for if/ admits 

 of considerable simplification, for, then we have ^ = oo , and the differen- 

 tial equation for i// becomes 



dij/ _ dz ^ zdz 



which on integration leads to 



log^= -log ^-i log (1 + 1 ^2;, 



or, xpz(l-^iz^)i = \, 



which, by putting z = tan 6, reduces to 



tan2(9 (2 + tan2^)' 

 The coordinates, therefore, of any point on the trajectory are given by 



X2= 1/^2 tan* (9 = 



Y2 = 4i//2 tan2(9 = 



2 + tan2 6l' 

 8X2 



2 + tan2^' 



which easily shew that the trajectory is the ellipse 



7/ + 2a-2 = 4A2. 

 Example IV. — To obtain the oblique trajectory of a pencil- of 

 coplanar rays radiating from a point, and whose equation is, therefore, 

 obtained by varying a in 



y = ax. 

 First Method. 



The coordinates of any point on the line are given by 



tj = aO, 



X = e, 



