472 
ON POLYZOMAL CURVES. 
[414 
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 con 
tinuously 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 Conics 
inscribed in a conic and touching three inscribed conics 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 Cremona, suggesting to him the problem solved 
in his letter of March 3, 1866, as mentioned in my paper, “Investigations in connexion 
with Casey’s Equation,” Quarterly Math. Journ. vol. Yin. 1867, pp. 334—341, [395], and 
as also appears, Annex No. IV of the present Memoir. 
In connexion 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*JW + GA V33° + ARV© 0 = 0 (where BC, CA, 
AB, denote the mutual distances of the points A, B, G) 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, h, c, f, g, ¡3, y) 2 = 0, where a, (3, y are circles, is a bicircular quartic. If we 
take the equation (a, h, c, f, g, hQX, g, v) 2 = 0 in tangential coordinates (that is, when 
\ g, v are perpendiculars let fall from the centres of a, /3, y on any line), it denotes 
a conic; denoting this conic by F, and the circle which cuts a, ¡3, 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 consequences, 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 Academy, 10th February, 1867, and published 
in their Proceedings, pp. 44, 45, contains the theorems mentioned in the letter of 
30th April, and some other theorems. It is not necessary that I should particularly 
explain in what manner the present Memoir has been, in the course of writing it, 
added to or altered in consequence of the information which I have thus had of 
Mr Casey’s researches; it is enough to say that I have freely availed myself of such 
information, and that there is no question as to Mr Casey’s priority in anything which 
there may be in common in his memoir on Bicircular Quartics and in the present 
Memoir.