52 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



as the envelope of the variable circle 



(x — acsmD 2 + y 2 = 6 2 cos 2 d , 



viz., of a circle having its centre on the major axis at a distance = ae sin 6 from 

 the centre, and its radius = b cos 0. (I notice, in passing, that this gives in 

 practice a very convenient graphical construction of the ellipse.) It may be re- 

 marked that for 6 = ± sin ~ ' e, the circle becomes 





viz., this is the circle of curvature at one or other of the extremities of the major 

 axis ; as 6 passes from to ± sin _1 e we have a series of real circles, which, 

 by their continued intersection, generate the ellipse; as increases from 

 = dzsin ~ x e to db 90 , the circles continue real, but the consecutive circles no 

 longer intersect in any real point, — and ultimately for = ± 90°, the circles be- 

 come evanescent at the two foci respectively. 



121. In the case q^> 1, we have a real representation of 



(x-qac)* + y* + b\q* -1) , 



as the squared distance of the point {x, y) from a point (X, 0, Z) out of the plane 

 of the figure, viz., putting this 



= (x - Xf + f- + Z* , 

 we have 



qae = X, Z 2 = b 2 (q* - 1) , 



whence 



* - <£ - ■ 



or what is the same thing, 



X 2 _ Z 2 _ . . 



a 2 - b 2 b 2 



that is, the locus is the focal hyperbola, viz., a hyperbola in the plane of zx, 

 having its vertices at the foci, and its foci at the vertices of the ellipse. 



122. If instead of the form first considered, we start from the trizomal form 



2bz + s/tf + ( y _ aeizf + J a? + (y + aeizf = , 



then we have the zomal or circle of double contact under the form 



x 2 + (y — qaci) 2 = a 2 (l — q 2 ) ; 



or putting herein q — — itan<£, this is, 



x 2 + (y — aetanp) 2 = a 2 sec 2 <p ; 



so that we have the ellipse as the envelope of a variable circle having its centre 

 on the minor axis of the ellipse, distance from the centre = aetancp, and radius 



