1881.] of Cartesians and other Quartics. 89 



10. When two foci B, G of the three foci A, B, G of a system 

 of confocal Cartesians coincide, then k=l, and we obtain the 

 systems of confocal limagons, 



r cosh 2f — r'= a) 



r cos 2?7 + r'= a) ' ■ 



where r is the distance of a point P from the double focus 0, r 

 from the single focus A, and a is the distance between the foci. 



Denoting the angle POA by 6, then 



r' 2 = r 2 — 2ar cos + a 2 , 



and we obtain the polar equations of the limagons 



r sinh 2 2£ = 2a (cosh 2£ - cos 6) 

 rsin 2 2?7 = 2a (cos — cos 2t})\ 



If the foci B, 0, after coincidence, now move at right angles to 

 the original line of foci, so that ABC is an isosceles triangle, we 

 obtain the system of curves derived from the integral 



dz 



/ 



^l{z-a.{z-mf-\-n i \ ' 

 1 — en u 



giving z — a=a- , 



& ° 1+Clltt 



where a = *J{ (m - a) 2 + n 2 } = A B or A C, 



and k 2 = I (l - - — ) = sin 2 i^ C 



If r r 2 , r 3 denote the distances of a point from the three foci 

 A, B, C, then 



1 — en (£ + wf) 1 — en (| — irf) 



r" = a 



or j\ = a 



1 en it} + en j 



Also z— m — in 



1 + en (| + irj) 1 + cn(£ - irj) 



/en t^ — en fY 

 \cn ir} + en %) 



en it; — en £ 



1 — en u 



= a — ra — in + a -= — 



1 + en w 



/ „ . ■ ~ n 1 — en u\ 

 = a(cos20-*sin20 + .,— 



\ 1 + en <// 



