PROFESSOR CAYLEY ON POLYZOMAL CURVES. 53 



= aseccp. This is, in fact, Gergonne's theorem, according to which the ellipse is 

 the secondary caustic or orthogonal trajectory of rays issuing from a point and 

 refracted at a right line into a rarer medium. It is to be remarked that for 



tan (p — ± -y , the equation of the circle is 



viz., this is the circle of curvature at one or other extremitity of the minor axis; 



etc 



from <p = to (p = ±tan _1 j , the intersections of the consecutive circles are 



real, and give the entire real ellipse; from <p = ±tan t-, to (p = ±90°, the 



circles are still real, but the intersections of consecutive circles are imagi- 

 nary. 



123. If in the equation of the generating circle we interchange cc, y, a, b, the 

 equation becomes 



(x — aei tan p) 2 + y 2 = b 2 sec 2 tp , 



which is (as it should be) equivalent to the former equation 



(x — aesind) 2 + y l — b 2 cos6 , 



the identity being established by means of the equation 



cos0 = , and .\ sin& = itai\<p, tan 6 = isine , 



cosp 



which is Jacobi's imaginary transformation in the theory of Elliptic Functions. 



Foci of the Circular Cubic and the Bicircular Quartic — Art. Nos. 124 to 126. 



124. For a cubic curve, the class is in general = 6, and the number of the 

 foci is = 36. But a specially interesting case is that of a circular cubic, viz., a 

 cubic passing through each of the circular points at infinity. Here, at each of 

 the circular points at infinity, the tangent at this point reckons twice among 

 the tangents to the curve from the point; the number of the remaining 

 tangents is thus = 4, and the number of the foci is = 16. If from any two 

 points whatever on the curve tangents be drawn to the curve, then the two 

 pencils of tangents are, and that in four different ways, homologous to each 

 other, viz., if the tangents of the first pencil are (1, 2, 3, 4), and those of the 

 second pencil, taken in a proper order, are (1', 2', 3 r , 4'), then we have (1, 2, 3, 4) 

 homologous with each of the arrangements (T, 2', 3', 4'), (2 7 , 1', 4', 3'), (3', 4', 1', 2'), 



VOL. XXV. PART I. 



