(18) 



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



property of these f anctions that the two curves 



cos X cosh y = a 

 sin X sinh y =b 

 intersect orthogonally at every common point of intersection. 

 It may similarly be shewn that if we have 

 r^ = n{\-0), M = 0, 

 the functions /^ and f^ are conjugate with respect to 6 and ij/ ; for the 

 above investigation remains unaltered, except in that we have 



L = - N, 

 so that (16) becomes 



dA_ __df, 



d^\, dO' 



and we have 



\de} dd d^' 



whence 



dO d^' 



and, by virtue of (18) and (19), it is again manifest that /^ and f^ are 

 two conjugate functions of and ij/. Consequently, as in the second 

 example given above, we have 



^=n (X-(9), M = 0, 



the quantities 



Ji cosh cos \|/, h sinh sin \f/ 

 are two conjugate functions of and ij/, and, the curves 



cosh X cos y = a 



sinh X sin y = h 

 are orthogonal tmjectories of each other. 



Again, it is an elementary principle in the theory of conjugate 

 functions that if <p and \j/ are any two conjugate functions of x and y ; 

 and if $, rj are any two other conjugate functions of x and y : then, by 

 putting i and rj instead of x and y in the values of <p and i|/, we get two 

 new conjugate functions of x and y. But, we have found above two 

 pairs of such functions, viz., 



(19) 



</> = sin X sinh y 

 \j/ = cos X cosh y 



1 



i = cosh X cos y "^ 

 7} = sinh X sin y j . 



