120 A. Mukhopadhyay — Differential Equation of a Trajectory. [No. 1,1887. 



= i a^ cosh xp sinh \p — ^ afiij/ 

 = ^ ON. NM - i aj8./r = ONM - f a)8./r 

 .'.0CM. = la/3xl/ = 0CB. if/. 

 But, ex }iy;pothesi, OCM = OCB. n(X-{-<{>) 

 .-.,/, = ^(X+(/>), 

 which shews that the co-ordinates of M are given by 

 a?! = a cosh if/=h cos <p. cosh n(\-{-(l>) 

 y^=.p sinh \p — h sin ^. sinh n(X-l-'?)> 

 and, therefore, M is a point on the trajectory. We thus see that not 

 only are the co-ordinates of M expressible in a very simple form, but 

 also that the position of M can be determined geometrically, correspond- 

 ing to any position of B on the circle ; hence, the curve can be com- 

 pletely traced. It is easy to remark that whatever may be the value 

 of the arbitrary constant X, the point M lies on the hyperbola CM, 

 for a given value of </>. Finally, a geometrical relation is worth notic- 

 ing, viz., since the circular sector AOB = ^h'^'p, we have from 



OCM = w(A+W OBC, 

 the equation 



0CM = .X0BC + 2.^^^-^^^ 



or, 0CM = nX0BC+2n 



OAB OBC 



OA OB 



