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414] ON POLYZOMAL CUftVES. 
viz., this is the circle of curvature at one or other extremity of the minor axis; from 
<f) = 0 to (p = + tan -1 , the intersections of the consecutive circles are real, and give 
the entire real ellipse ; from </> = + tan -1 to <£ = + 90°, the circles are still real, but 
the intersections of consecutive circles are imaginary. 
123. If in the equation of the generating circle we interchange x, y, a, b, the 
equation becomes 
(x — aei tan </>) 2 + y- = & 2 sec 2 </>, 
which is (as it should be) equivalent to the former equation 
(x — ae sin 0) 2 + y- = b° cos 6, 
the identity being established by means of the equation 
cos 6 = r, and therefore sin 6 = i tan <6, tan 6 = i sin <6, 
COS (f) 
which is Jacobi’s imaginary transformation in the theory of Elliptic Functions. 
Article Nos. 124 to 126. Foci of the Circular Cubic and the Bicircular Quartic. 
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', 4'), 
then we have (1, 2, 3, 4) homologous with each of the arrangements (1', 2', 3', 4'), 
(2', 1', 4', 3'), (3', 4', 1', 2'), (4', 3', 2', 1'). And in each case the intersections of the 
four corresponding tangents lie on a conic passing through the two given points on 
the curve ( x ). 
1 It may be remarked that if the equation of the first pencil of lines be 
{x-ay)(x- by) (x - cy) (x - dy)=0, 
and that of the second pencil 
[z - aw) (z - bw) (z-cw) (z—dw)=0, 
then the equations of four conics are 
xw-yz — 0, 
(a + d-b-c)xz + (bc-ad) (xw + yz) + (ad(b + c)-bc (a + d))yw=0, 
(b + d - c - a) xz + (ca - bd) (xw + yz) + (bd (c + a) - ca (b + d)) yw=0, 
(c + d - a - b) xz + (ab - cd) (xw + yz) + (cd (a + b) - ab (c + d) ) yw=0. 
C. VI. 
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