1888.] A. Mukhopadhyay — Differential Equations of Trajectories. 85 

 and these values sliew that 



M = 



]sr= -4 (^2+^2). 



The differential equation for if/, consequently, becomes 



whence 



i|/ = w (\ — 6). 

 Hence, finally, the coordinates of any point on the trajectory are given by 



x = e^-n^ (x-oy. 



Y=:2n0{\-e). 

 Since X and T are two quadratic functions of the parameter 0, it is clear 

 that the trajectory must be a conic ; in fact, the actual equation is 



(l^-7^2)^ (a;S + 2/2)= ^(n^ -I) x + 2mj -2n^X^y, 



which may be thrown into the form 



\^2nx - {n^ - I) yY = 4^n^ \^ [ 7i^ AS _ (^2 - 1) ^ _ 2ny } , 



which shews that the trajectory is a parabola, and, if n = tan a, the polar 

 equation is 



\/r. sin f a+ - J = A. sin a. 



Example VII. — To find the oblique trajectory of the system of 

 curves obtained by varying h in the equation 



e"^ sin y = ah. 

 The coordinates of any point on the curve are given by 



x = \og a \/¥Th^ 



2/ = tan'^g. 



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



X = log a \/W^ 



Y = tan-^|. 

 u 



To determine i// as a function of B, we have 



/i = loga + ilog(^3_i.^2), 



/2 = tan-^i 



which give the values 



Af, 6 



dS 68++= 



