1888.] A. Mukhopadhjay — Blfferential Equations of Trajectories. 89 



Heuce we have the two new conjugate functions 



sin j cosh x cos y \ sinh | sinh x sin y \ , 



cos { cosh X cos y > cosh | sinh x sin y > . 

 We have, therefore, the theorem that the two transcendental curves 

 sin I cosh x cos y \ sinh | sinh x sin y > =a 



cos I cosh X cos y I cosh J sinh x sin ?/ | = 6 



are orthogonal trajectories of each other. In the same manner, it may 

 be shewn that the quantities which furnish the coordinates of any point 

 on the trajectory in terms of 6 aud if/, in the second method of establish- 

 ing Example IV, as well as in Examples VI and VII, are conjugate 

 functions. 



We shall now give some examples in which the properties of conju- 

 gate functions will materially simplify the calculation. 



Example VIII. — Consider the tricircular sextic 



(^2 + 2/2) (^2 4- ,^2 _^ 7,2)2^^2 |^ ^3 (^2 + ^2 _ ^2)^,^2 (^24-7/4- A;^)^ ] , 



and suppose that its oblique trajectory is required when a is made to 

 vary. Writing 



a^=l + h% 

 the equation may easily be thrown into the form 



^2 (^.2 4-^2+7,2)2^^2^,2 (^2 + ,^2_/„g)2_^^2.^3 (^.2 + ^2 + 7^2)2^ 

 whence it can bo shewn without much difficulty that the coordinates of 

 any point on the sextic curve are given by the system 

 ^2 a — cos 6 62 



^ ~ a + cos 0' 62 + sin2 ' 

 ?/2 a — cos sin^ 

 P~a + cos 0' 62 + sin2^' 

 and we seek the oblique trajectory, when a and h are made to vary 

 subject to the conditions 



a = cosh \p, 

 h =z sinh \\). 

 The coordinates of any point on the trajectory are given by 

 X2 cosh xp — cos sinh^ i|/ 



P cosh i|/ 4- cos 0' sinh2 rp -]- sin2 

 Y2 _ cosh yp, — cos 6 sin2 



A;2 "~ cosh ij/ + cos 6' sinh''* ij/ ^ sin'-^ 

 To determine 4/ as a function of 0, we have 



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