82 



Mr Greenhill, On Conjugate Functions [Mar. 21, 



Therefore r,, r 2 , r 3 , r it r denoting the distances of a point 

 from the foci and from the origin, 



r 2 = sn (£ + iy) sn (| — irj), 



rj\ = en (| + m?) en (f - 117), 



r i r 4 = p dn (£ + in) dn (£ - m;); 



and therefore writing in equations (A) r 2 for r, r 2 r 3 for r , and 

 r^ for r", then 



r 2 r 3 — ? ,2 dn2£ = cn2£ ) 



or 



or 



?\r 4 — r 2 cn 2£ = p dn 2| i 



> . 



r x r 4 4- r 2 cn 2^ = p dn lit] \ 



k j 



&' 2 l 



r x r 4 dn 2£ - r 2 r 3 cn 2£ = p- | 



W* 



r 1 r i dn 2177 — r 2 r 3 cn 2 177 = p- I 



A/ J 



represent one and the same system of orthogonal quartic curves. 



5. For x + ly = cn (f 4- i^), 



the foci are given in order by 



.h' 



z = 1 > i 1> -li 



'Jb ' 



and 



r 2 = cn (£ + it]) cn (£ — ^77), 

 r 1 r 3 = sn(^ + Z7 ? )sn(^-i77), 



V4 = p dn (f + z't?) dn (£ - i v ) 



and therefore the quartic curves are given by 

 r 2 — rj\ dn 2£ = cn 2f 



?^ 2 + 7\r s dn 2z*77 = cn 2it]) ' 

 with similar expressions connecting 



r 2 and r 2 r 4 , r^ and r 2 r 



