TRANSACTIONS OF SECTION A. 461 



From these equations we get 



13. If we substitute -^-*-4 for/($) in art. (5), equation (9), we get 



(20) 



,= { 1 + (4>) 2 / FW,) - 



Hence, if p = F((/>) be the polar equation of a curve, the intrinsic equation of 

 the evolute of its first negative pedal is 



-{ i+ U)'} F( * ) - 



In like manner, from art. (8) the intrinsic equation of the involute of its nega- 

 tive 'pedal is 



PARALLEL CURVES. 



14. If v = f(p be the tangential equation of a curve, we have at once the 

 tangential equation of a parallel curve at the distance k given by the equation 



v =f(j> ± k cosec <£. (21) 



This equation enables us to write out at once the equation of the reciprocal of 

 the parallel curve, which is evidently the curve whose polar equation is 



%■ =/(<£) sin <p±k, 

 P 



Where y is the radius of the circle of reciprocation. Thus the equation of the 

 reciprocal of the parallel to the ellipse 



v = */a' + b~ cot* <p. 



2 



is — = V (« 2 sin 2 <p + b* cos 2 + k. 



or in Cartesian co-ordinates 



4r 4 /,; 3 (a* + y 2 ) = («V) + - k\x 2 + y 2 ) V. (22) 



SECTION III. 



Rectification of Bicircular Quartics. 



(15.) Being given a conic F (called the focal conic) and a circle J (called the 

 circle of inversion), it is known that a bicircular quartic is the envelope of a 

 variable circle whose centre moves on F, and which cuts J orthogonally. It is 

 proved in my memoir on " Bicircular Quartics " that there is a fourfold generation 

 of the curve, -viz., there are four focal conies, F, F', F", F'", and these are con- 

 focal, and the corresponding circles of inversion J, J', J", J'", are mutually ortho- 

 gonal. The rectification of the quartic depends on the following geometrical 

 theorem. In a bicircular quartic there exists a series of inscribed quadrilaterals 

 ABCD, whereof the sides AB, BO, CD, DA pass through the centres of the four 

 circles J, J', J", J"', respectively ; or, as it may be expressed, the pairs of points 

 (A, B), (B, C), (C, D), (D, A) belong respectively to the four modes of generation. 

 Now consider the quadrilateral ABCD, and giving it an infinitesimal variation, we 

 have four infinitesimal arcs, AA', BB', CO', DD', which we shall denote by ds, 

 ds% ds", ds'", respectively ; now let the radii of the four generating circles which 

 touch the quartic at the four pairs of points (A, B), (B, 0), (C, D), (D, A) be p, p', 

 p", p'", respectively. 



