62 Proceedings of the Royal Irish Academy, 



yil. — On the Geometeical Properties oe the '' Atriphthaloid." 

 By the Rev. Richaud Townsend, M. A., F.T.C.D. 



[Read, February 27, 1882.] 



Dr. HattghtojST has discovered the existence of a large family of curves 

 in the course of his investigation of the form of a frictionless ocean, 

 covering an attracting sphere. 



The whole family he proposes to call Atriptothalassic curves, but 

 one of them is so simj^le and elegant, that he has named it the 

 Atriphthaloid, and requested me to undertake its discussion, which 

 I have done in the following Paper : — 



The equation of the AtripMhaloid in polar co-ordinates is 



r'(-4 - 2gr) = c^ cosec'6. 



Putting for convenience of discussion, A = 2gh, c- = 2yk'^, and 6 - 90° - w, 

 the equation of the curve assumes the form — 



r\r-h)+Fsec'w = 0, (1) 



where h and Jc are positive constants representing linear magnitudes, 

 which may have any independent values from to cc, and which may 

 he regarded as the parameters of the curve. 



The equation (1) giving the same values for r when w is changed 

 into - (0, or into tt ± w, we see at once that the curve it represents is 

 symmetrical with respect to the two rectangular axes w = and w = ^ tt, 

 and has a centre at the origin (see fig.) 



The equation (1) being, for every value of w, a cubic in r whose 

 absolute term is positive : hence, for every value of co, r has one real 

 negative value, commencing from its minimum absolute value OC 

 when (D = 0, and increasing continuously to oo from co = to w = ^tt. 

 Hence (see fig.), the two symmetrical conchoidal-shaped infinite 

 branches, which always meet asymptotically on the axis for which 

 (J} z= ^ TT, and which never disappear for any finite values of h and k 

 however related to each other. 



The remaining two roots of the cubic ( 1 ) for r in terms of .w being 



real or imaginary, by the theory of equations, according as sec'-^ w is < 



4 7r 4 h^ 



or> ^ ,3; hence, when — y^ is > 1, the curve (see fig.), in addition 



to the two infinite symmetrical conchoidal branches which never dis- 

 appear, has two finite symmetrical ovals, lying entl^-ely outside the 

 eonchoidal branches, and intersecting the axis of x at two pairs of real 



