90 A. Mukhopadliyay — Differential Eqiiations of Trajectories. [No. 1, 



and then by actually calculating* the values of 

 ^ dj. df, d£^ 



dO' tie' d^\,' d^' 



we can shew that 



L4-N = 0, M = 0, 

 whence the differential equation for ^^ becomes 



d^ 



and ^z=. n {\ — 6). 



But, from the theorem we have established at the beginning of this 

 section, we know that the same conclusions may be legitimately drawn 

 without direct calculation, if we can prove /^ and f^ to be two conjugate 

 functions^ and we proceed to do so. Now we know that if 



tanj {6-\-s/'^l ;|;) = A + \/^B, 

 the two conjugate functions A and B are given by the system 



A2 _ sin^ e 

 B2 ~ sinh2 xf, 

 ^9. I gg_ cosh;{/-cosg 

 cosh ij/ + cos 6'' 

 whence it follows that 



cosh i|/ — cos 6 sin® 



A2 = 

 B2 



cosh Tj/ + cos 0' sinh2 ^ ^ gin^ 

 cosh 1^ — cos 6 sinh^ \(/ 



cosh i(/ + cos 0' sinh^ \f/ + sin^ 6 

 But these are the quantities which when multiplied by h^ reproduce the 

 squares of what we have called / j and f^ above, which was to be proved. 

 Hence we finally infer that the coordinates of any point on the sextic 



may be represented by the equations 



x^ a — cos 6 b^ 



k^ a-l-cos 0' 62_|-sixi2 0^ 

 y^ a — cos 6 sin^ 

 ^ ~ a + cos 0' 62-|_sin2 (9' 

 where a^ — 6^ = 1, 



and, accordingly, the coordinates of any point on its oblique trajectory 

 are furnished by the system 



X^ _ cosh. n(X-O) -cos sinh^ n(\ - 6) 



1^~ Goshn(X-e)-j- cos6' sinh2 n(\-e)+sin^ 0' 

 Y* _ cosh n(\ -6)- cos 6 sin^ 



'W " cosh n{k-e)^ cos 6' sinh^ w(A- 6>) -f sin^ ^* 



