TWEMY-SfXTlI ANNUAL MEETING. . 15 



that is, tlie four lines reduce to two pairs of coincident lines, showing that the 

 line at infinity cuts the curve in two imaginary double points. 



The curve has four foci, two real and two imaginary: for the foci of a curve 

 are determined by the points of intersection of tangents from the circular points 

 at intinity. In the case of a curve of the foui-th degree there are in general twelve 

 such tangents ; but as the circular points at infinity are also double points, there 

 are in this case but four distinct tangents, which intersect in two rcsal and two 

 imaginary points. The two real points are the foci F, F' mentioned above. 



When 6:^0, a = b, and the equation breaks up into the two factors, 



x^-\-y^z=a^, and x^+p^=0; (10) 



that is, the curve is a circle of radius a and a point circle at the origin. When 

 e > and < y'Y^ the curve is a smooth oval. When e = |/|~, it changes to an 

 indented oval. In order to determine when this change occurs, it is only neces- 

 sary to express the condition that the curvature at the x axis shall be zero. Using 

 the polar equation 



r^= a- eos^ k + h- sin^ k, (11) 



the initial line being the axis of abscissEe, the condition for zero curvature at a 

 point is : 



r^+2 — ? =0, (12 



which becomes, when values are substituted : 



a^ cos^ k + &« sin^ k — {h^ — a^) cos 2 k (13) 



+ I (68 — a*)* sin^ 2 k (a^ cos^ k-{-b^ sin^k)-^ = 



When k = this reduces to Sa^ = b^, whence e = v/^, the required condition. 



When e > y' 1" and e < 1, the curve is an indented oval, with four real points 



of inflection. In order to determine these points of inflection, set 



d^u , „ „ ,, 



— + „ = ». (14, 



where « ^ — of equation (11). 



This gives, after substituting values and reducing, 



i 



(15) 



— 1 , « 



k^fan ± — 



b 



b' — 2a^ 



2b-^—a'' 



Equation (15) shows that there are four points of inflection symmetrically 

 placed about the origin, as might have been inferred from the form of the curve. 

 Substituting the values of sin^ k and cos- k obtained from equation (15), in equa- 

 tion (11), the radius vector to the points of inflection is found to be 



r = — (16) 



V2 Va-+ &2 

 showing that the four points of inflection lie on a circle concentric with the curve. 

 When e = 0; b= a and tan k is imaginary; showing that there are no real 

 points of inflection. 



When e = -/J; 6^ = 2a'', tan k = o, and r = ± a, showing that the four points 

 of inflection coincide two and two at the extremities of the minor axis. This is the 

 transition from a smooth to an indented oval. When e = l; a = 0, tan k = 0,r 

 = 0; that is, the points of inflection are all at the origin. 

 In this case the equation of the curve breaks up into 



a-2 + 2/2 = 6v/ and x'^ + ^2 ^ — by, (17) 



two circles of radius lb, tangent to the axis of abscissa at the origin. They will 

 be finite if e becomes unity by a being zero; infinite, and hence simply two 



