78 Mr Greenhill, On Conjugate Functions [Mar. 21, 



and therefore 



r + r — cos 2irj = cosh 2 77, 



r —r= cos 2£ ; 



giving the confocal ellipses and hyperbolas. 



The Jacobian of the system 7 ' = f (£ + irj) /" (£ — irj) 

 if z=f(% + irj), and is therefore equal to 



4 sin (| + irj) cos (£ + ^77) sin (f — i??) cos (£ — irj) or 4rr'. 



2. Now if we consider the integral 



f rf^ „ 



J V(^- 1-s.l-F*) " 



then 2 = sn 2 w, 



or oc + iy = sn 2 (£ + irj) ; 



and if r, r, r" denote the distances of a point from the three foci 



z = 0, 1 and p, then 



r = sn (£ + 117) sn (f — 177), 

 ?•' = en (| + irj) en (£ — {77), 



r" = p dn (| + irj) dn (f? - irj) ; 



and we easily obtain 



r'-rdn2£=cn2f, 



r en (277, k') + r dn (2?;, h') = 1 ; 



so that the curves £ = constant, and rj = constant are confocal 

 Cartesians, cutting at right angles by reason of the fundamental 

 property of conjugate functions ; which is Prof. Crofton's theorem. 



For r = sn (£ + irj) sn (£ — irj), 



_ en 2irj — en 2£ 

 "* dn 2ir) + dn 2f 



_ 1 dn 2ir) - dn 2g 

 ~ F en 2t"?7 + en 2| ' 



