76 A. Mukhopadhyay — Differential Equations of Trajectories. [No. 1, 



while the relation between a and h given in (13), is equivalent to 



a—h cosli \l/, 

 b = h sinh i//, 

 so that, the coordinates of any point on the trajectory are given by 



X = 7i cos ^ cosh i/^, (14) 



y =:/i sin ^ sinh i/a (15) 



Again, to determine the relation between 6 and i}/, we have 



fi — Ti cos 6 cosh i/^, 



/g = h sin 6 sinh i/^, 

 which lead to the system 



-^ = — i^ sin ^ cosh x^/, 

 du 



-— = h cos 6 sinh i/^, 

 do 



-7-7 = h cos ^ sinh ^, 

 d\p 



~ = ^ sin cosh i^, 



and, these give 



L = 7^2 (sm2 e cosh' i/^ + cos^ e sinh^ »/.), 



M = 0, 



N = 7^2 (sin2 61 cosh2 ,/.+ eos^ (9 sinh^ ,/,), 

 so that, the differential equation (11) becomes 



d^b 



whence, \p =:n {X-\-0), 



where A. is the constant of integration. Substituting in (14) and (15), 

 we see finally that the coordinates of any point on the oblique trajectory 

 of a system of confocal ellipses, are given by 



'K = h cos cosh 71 (A+ 6)j 



Y = hsine sinh n (\ + 0), 

 which is exactly the system of equations to which Mainardi's result was 

 reduced in my former paper, and geometrically interpreted there. 



§. 4. Other applications of the Theorem. 



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

 confocal hyperbolas 



where 



a^ 



62- 



1, 



a2 



+ 62:= 



h^ 



