PROFESSOR CAYLEY ON POLYZOMAL CURVES. 3 



WA 5 + \/roB° + \Sn(y = 0, and the tetrazomal curve VIK U + VmB° + VnC° + VpD° 

 = 0, to which the concluding portions relate. I have accordingly divided the 

 Memoir into four parts, viz., these are— Part I , On Polyzomal Curves in general ; 

 Part II., Subsidiary Investigations ; Part III., On the Theory of Foci ; and Part IV., 

 On the Trizomal and Tetrazomal Curves where the zomals are circles. There 

 is, however, some necessary intermixture of the theories treated of, and the 

 arrangement will appear more in detail from the headings of the several articles. 

 The paragraphs are numbered continuously through the Memoir. There are 

 four Annexes, relating to questions which it seemed to me more convenient to 

 treat of thus separately. 



It is right that I should explain the very great extent to which, in the com- 

 position of the present Memoir, I am indebted to Mr Casey's researches. His 

 Paper " On the Equations and Properties (1.) of the System of Circles touching 

 three circles in a plane; (2.) of the System of Spheres touching four spheres in 

 space ; (3.) of the System of Circles touching three circles on a sphere ; (4.) on 

 the System of Conies inscribed in a conic and touching three inscribed conies in 

 a plane," was read to the Royal Irish Academy, April 9, 1866, and is published 

 in their " Proceedings." The fundamental theorem for the equation of the pairs 

 of circles touching three given circles was, previous to the publication of the 

 paper, mentioned to me by Dr Salmon, and I communicated it to Professor Cre- 

 mona, suggesting to him the problem solved in his letter of March 3, 1866, as men- 

 tioned in my paper, " Investigations in connection with Casey's Equation," 

 " Quarterly Math. Journal," t. viii. 1867, pp. 334-341, and as also appears. 

 Annex No. IV. of the present Memoir. 



In connection with this theorem, I communicated to Mr Casey, in March or 

 April 1867, the theorem No. 164 of the present Memoir, that for any three given 

 circles, centres A, B, C, the equation BC */A° + CA\^B° + ABVC° = (where 

 BC, CA, AB, denote the mutual distances of the points A, B, C) belongs to a 

 Cartesian. Mr Casey, in a letter to me dated 30th April 1867, informed me of 

 his own mode of viewing the question as follows : — " The general equation of the 

 second order (a, b, c, /, g, h) (a, ft, y) 2 = 0, where a, ft, y are circles, is a bicircular 

 quartic. If we take the equation (a, b, c,f, g, h) (X, p., vf = in tangential co-ordi- 

 nates (that is, when \ n, v are perpendiculars let fall from the centres of a, ft, y 

 on any line), it denotes a conic ; denoting this conic by F, and the circle which 

 cuts a, ft, y orthogonally by J, I proved that, if a variable circle moves with its 

 centre on F, and if it cuts J orthogonally, its envelope will be the bicircular 

 quartic whose equation is that written down above;" and among other conse- 

 quences, he mentions that the foci of F are the double foci of the quartic, and the 

 points in which J cuts F single foci of the quartic, and also the theorem which I 

 had sent him as to the Cartesian, and he refers to his Memoir on Bicircular Quartics 

 as then nearly finished. An Abstract of the Memoir as read before the Royal Irish 



