TowNSEND — Geometrical Properties of the Atriphthaloid. 65 



semi-axes a, h, c of the curve, that is of the three distances OA, OB, 

 (9 C of the three vertices A, B, C from the origin 0, the cubic equation 



x^ - AV + 2hl^x- -/!;«= 0, (4) 



which has always one real positive root, and whose three roots when 

 real are all positive. 



N.B. — The equation (4) is, by the theory of equations, that whose 

 roots are the squares of those of equation (1) for the case when w = 0. 



Since from equation (1) when w = 0, by the theory of equations. 



a + I + G = h, 



he + ca -\- ah = 0, 



ahc - - ]^ 



ah 



or + ah +h'^ 



"a + h' 



(5) 



r f 



and if d denote the distance 01) (see fig.); since (P'=3 —, therefore also 



a-b'^ 



d^= 3 — = — ; relations which give the values of the four quanti- 



(f + ah-v¥ ^ 



ties c and d, h and ]c, in terms of a and I, and show consequently that 



the two latter quantities, which when real may have any independent 



values from to go, determine completely all the particulars of the 



curve in every case. 



In the particular case when 4A^ = 2lJc', that is when the two ovals 



contract into points, and when consequently a = h, equation (4) has 



4 . . 1 



two of its roots each = -/r, and its third root =-h^. 

 9 9 



In the particular case when h = 2k, in which as 4A^ > 27X-^ the two 

 ovals are real, the three roots. of equation (4) are respectively x'^ = k', 

 andr' = i (3 ± V5) k-. 



In the same case, those of equation (1), when w - 0, are respec- 

 tively r = k, and r = ^ (1 ± V5)^; as they ought, the roots of equa- 

 tion (4) being in every case the squares of those of equation (1), when 

 in the latter w = 0. 



Putting in for h and k in the function (4A^ - 27k^) k^, on whose 

 sign as positive or negative it depends whether the ovals are real or 

 imaginaiy, their values in terms of a and b as given by equation (5), 

 we find readily that 



(U^ - 27 k')k^ = 



(2(fl-+ ah + h^)+ 3ah)ah 

 {a + bf 



{a - by, (6) 



and that its sign is consequently, as it ought to be, positive or nega- 

 tive according as a and b are real or imaginary. 



