92 A. Miikliopadhyay — Differential Equations of Trajectories, [No. 1. 



We see, therefore, finally that the coordinates of any point on the 

 oblique trajectory of the bicircular quartic 



which is obviously the inverse of an ellipse, may be represented by the 

 system 



2 cos 6 cosh n(X — 6) 



X = 



h [ cosh 2w(X-^)+ cos 2(9] 



_ 2 sin 6 sinh n{\ — 6) 



h [cosh2w(A-6')+ cos 2(9J 



when a and h vary subject to the relation 



7^2 = a^ - 62. 



Example X. — Again, if we seek the oblique trajectory of the 

 transcendental curve 



1% a^e-'^-li^e'' 



tan^ y 



(^^ h'e"^ 



when a and b vary subject to the condition 



aS - 62 _ 7^2^ 



we see that the coordinates of any point on the curve are given by the 

 system 



n^ J- 7i2 



2e2-=^-±i.-cos2^. 



cot 1/ — -r tan 6. 



But as 



a = h cosh i|/, 

 b = h sinh ^, 

 the coordinates of any point on the trajectory are given by 



, - cosh 2x1/ — cos 26 

 X = 4 log 2 



cot Y = coth \p tan 6. 

 To determine i)/ as a function of ^, we notice that the quantities 

 /, = I log 2 (cosh 24- - cos 26) - log 2 

 /g = cot"''' (coth ;|/ tan ^) 

 are two conjugate functions, being in fact exactly the two quantities 

 which we obtain in separating the real and imaginary parts of 



log sin (6-\-\/ — Ixp) 

 Hence, by the theorem of this section, we have 



L + N = 0,M=0, 



